1) The King

The Ancient Egyptians used several words to denote their king. In some cases, the exact meaning of a word referring to the king or its relationship to other, similar words is not known.

The most common word to refer to the king was , nsw, often abbreviated to . Note the graphical, or perhaps honorific transposition of the signs, whereby , sw, was moved to be the first sign of the group. The word nsw is derived from nj-sw.t, "the one who belongs to the sedge plant", with the sedge plant as the heraldic symbol for Upper Egypt. This word is therefore normally translated as "King of Upper Egypt", although its context often favours the shorter translation "King".

A word often used together with nsw was , bi.tj, "the one who belongs to the bee", where the nee is normally taken as a reference to Lower Egypt. bi.tj is thus translated as "King of Lower Egypt". Contrary to nsw, however, bi.tj is only rarely used in contexts where the shorter translation "King" should be favoured.

The combination of these two words results in   nsw-bi.tj, "King of Upper- and Lower-Egypt", or more literally "he who belongs to Upper-Egypt (the sedge plant) and Lower-Egypt (the bee)". This combination is part of the royal titulary and will be discussed below. It can, in some contexts, be translated simply as "King".

We commonly refer to the kings of Ancient Egypt as "Pharaohs". This was the word used by the Greeks and the Hebrews to denote the rulers of the Nile-country. It is derived from the Egyptian  pr aA, "the Great House", a word originally meaning "palace" or "court". 
From the end of the 12th Dynasty onwards the health wish "may it live, prosper and be in health" was often added when referring to "the Great House", but still it seems to mean only the palace or the court. 
The earliest certain instance where "the Great House" actually refers to the king is in a letter to Amenhotep IV (Akhenaten), which is addressed to "Pharaoh, may he live, prosper and be in health, the Master". 
From the 19th Dynasty onward pr aA is used occasionally to refer to the King instead of the palace. We read "Pharaoh did such and such".
The final development was when a proper name was added to the title, as in "Pharaoh Hophra". The earliest known Egyptian example of this use is under one of the Shoshenks of the 22nd Dynasty. 

All words related to royalty, including the names of the Pharaohs, could be followed by the auspicious wish-formula  "may he (or she or it) live, prosper and be in health". This could be carried to the extreme, where almost every word referring to a king might be followed by this wish.

2) The Royal Titulary

a) The Horus-name

The oldest known part of the royal titulary is the Horus-name   , sometimes also called the banner-name or the Ka-name. It represents the king as the earthly embodiment of the god Horus, the divine prototype and patron of the Egyptian kings. 
This name is ordinarily written within a rectangular frame, at the bottom of which is seen a design of recessed panelling, such as we find in the facades of early tombs and in the false doors of many private tombs. The Ancient Egyptian name for this facade was serekh. This name is often used in modern texts as well when speaking of the (palace) facade. 
On the top of the serekh is perched the falcon of Horus, hence the appellation "Horus-name". In more elaborate New Kingdom examples Horus is wearing the double crown and is accompanied by the sun and a uraeus. 
In the Early-Dynastic Period, the perched falcon of Horus was in fact part of the name of the king. Aha, for instance, was actually called Horus-Aha, "Horus who fights". 

This name was not the birth name of the king, but it was given to him when he ascended the throne. During the first three dynasties it was the king’s official name. His name of birth would not appear in official documents. This has complicated the identification of many early kings mentioned in the king lists, where only the name of birth is mentioned. 
Although it would continue to be used throughout the entire Ancient Egyptian history, it lost its importance to the Prenomen en nomen from the end of the Old Kingdom on. 

b) The Nebti-name

The Nebti-name   shows the king in a special relation to two goddesses: the vulture-goddess Nekhbet of the Upper Egyptian cities of Elkab and Hierakonpolis and the cobra-goddess Uto of the Lower Egyptian city Buto. Both goddesses are the deified personification of Upper- and Lower-Egypt respectively, and as such, the Nebti-name denotes the king as "the one of Nekhbet (Upper-Egypt) and Uto (Lower-Egypt)", i.e. as the "one belonging to Upper- and Lower-Egypt".  It is often translated as "Nebti" or "The Two Ladies".
1st Dynasty king Den is the first to have assumed this Nebti-name. The use of this title by Den may perhaps indicate some governmental reforms that may have occurred during this king's reign.

c) The "golden Horus name"

The meaning of the third part of the royal titulary, the "golden Horus name"    , is more disputed. It represents the falcon god Horus perched on a symbol that usually represents "gold". 
Based on the Greek equivalent of this title on the Rosetta Stone, which translates into English as "superior to (his) foes", it has been proposed that the hieroglyphs symbolised Horus as victorious over Seth, "the Ombite" (another possible reading of the hieroglyph on which the falcon is standing). This was, no doubt, the interpretation of Greek times, when the opposition between Horus and Seth was much more pronounced than in earlier times. For these earlier periods, however, the evidence may point in another direction. 
If the "golden Horus name" symbolised Horus’ victory over his enemy Seth, one might expect that the names following this group should be aggressive in nature, but most of the time, those names are far from being bellicose.
In a context dealing with the titulary of Thutmosis III that king says "he (Amun) modelled me as a falcon of gold". Thutmosis III’s co-regent Hatshepsut calls herself "the female Horus of fine gold". The concept of the golden falcon can be definitely traced back to the 11th Dynasty. An inscription of the 12th Dynasty describes the golden Horus name as the "name of gold". 
The notion of "gold" is strongly linked to the notion of "eternity". The burial chamber in the royal tombs of the New Kingdom was often called the "golden room", not (only) because it was stacked up with gold, but because it was there for eternity. The "golden Horus name" may convey the same notion of eternity, expressing the wish that the king may be an eternal Horus.

d) The Prenomen

The Prenomen is the name that follows the title    "King of Upper- and Lower-Egypt". The oldest known example of this title is again dated to the reign of 1st Dynasty king Den, when it was often combined with the Nebti-name, without a distinct name added to it. It would take until the end of the 3rd Dynasty before this title really came into use. It would, eventually, replace the Horus-name as most important official royal name.
The Prenomen itself almost always contained the name of the god Re. Typical examples are "pleasing to the heart of Re" (Amenemhat I) and "lord of the cosmic order is Re" (Amenhotep III). One of the first cases of Re as an element in a king’s name is with Khephren of the 4th Dynasty (Khaf-Re). 
The title "King of Upper- and Lower-Egypt" can sometimes be followed by the phrase   "the Lord of the Two Lands", which sometimes even replaces it entirely. A queen can be called "the Mistress of the Two Lands".

e) The nomen

The nomen is introduced by the epithet   "son of Re". It was added to the royal titulary in the beginning of the 4th Dynasty. It was from that time on that the royal titulary became established in the form discussed here. 
The name following this title was, as a rule, the king’s name of birth. It is almost the equivalent of our family name, for the 11th Dynasty affect the names Antef and Mentuhotep, the 12th Dynasty the names Amenemhat and Sesostris, the 13th Dynasty shows several kings of the name Sebekhotep and the 18th Dynasty consists almost entirely of ruler named Amenhotep or Thutmosis.
Sometimes, the phrase   "the good god" is placed before the nomen of the king, in addition to or in place of the "son of Re". Another title sometimes placed between "son of Re" and the actual nomen was   "lord of the apparitions", sometimes also translated as "lord of the crowns". This title again confirms the narrow link between the king and the sun: the king’s apparition on the throne is compared to the rising of the sun on the Eastern horizon. 

From the later half of the Old Kingdom on, the principal name is the Prenomen, and this is often found alone or accompanied only by the nomen. The Horus-name would serve only rarely for identification purposes. 
Both Prenomen and nomen are almost invariably written within "cartouches" or "royal rings". The cartouche depicts a loop formed by a rope, the ends tied together so as to offer to the spectator the appearance of a straight line:  . Strictly speaking this loop should be round as it conveys the notions of "eternity" and "encompassing the entire creation". It is elongated and oval because of the length of the hieroglyphic names enclosed in it.  In transcription, it is a good practice to indicate a cartouche by parenthesis: (ppy), Pepi.
Occasionally, one may find the name of a god or goddess in a cartouche. This was especially the case for Osiris-Onnophris and Isis in the temple inscriptions of the Greek-Roman Period. 

Cartouches were often followed by either a wish-formula such as  "living eternally" or  "bestowed with eternal life", or by a phrase which relates the king to a particular deity, for instance  "beloved by Amun-Re". Note the honorific transposition whereby the name of Amun-Re was moved to the start of the phrase.

3) Some royal names

We will now proceed with a list of prenomen and nomen of some of the most popular kings. It is possible that you do not know all the signs, but it should be possible to derive the value of some unknown signs through the transcription that is provided. It should be noted that for most, if not all names provided here, there are several variants.

Cheops (Khufu)

xwfw
Khufu (Greek version: Cheops).
This name is an abbreviation for Xnmw-xw=f, "Khnum is his protector".

Chefren (Khafre)

xa=f ra
Khafre (Greek version: Chefren).
This name means "Re appears".

Mykerinos (Menkaure)

mn-kA.w-ra
Menkaure (Greek version: Mykerinos)
This name means "The Kas of Re remain". Note that repeating a sign three times indicates a plural, which, in transcription is rendered as .w.

Neferirkare

nsw-bi.tj (nfr-ir-kA-ra) sA ra (kAkAi)
The King of Upper and Lower Egypt (Neferirkare), the son of Re (Kakai).
Neferirkare means "It is good what the Ka of Re has done". Kakai is probably the king's personal name. Neferirkare is the oldest known king to have had a prenomen and a nomen.

Unas

wnis
Unas.

Pepi I

nsw-bi.tj (mry-ra) sA ra (ppj)
The King of Upper and Lower Egypt (Merire), the son of Re (Pepi)
Merire means "Beloved of Re".

Pepi II

nsw-bi.tj (nfr-kA-ra) sA ra (ppj)
The King of Upper and Lower Egypt (Neferkare), the son of Re (Pepi)
Neferkare means "The Ka of Re is beautiful".

Mentuhotep II

nsw-bi.tj (nb-xrw-ra) sA ra (mnTw-Htp)
The King of Upper and Lower Egypt (Nebkherure), the son of Re (Mentuhotep)
Nebkherure means "Re is the master of the voice". Mentuhotep means "(the war-god) Mentu is at peace".

Amenemhat I

nsw-bi.tj (sHtp-ib-ra) sA ra (imn-m-HA.t)
The King of Upper and Lower Egypt (Sehetepibre), the son of Re (Amenemhat)
Sehetepibre means "The one who appeases the heart of Re". Amenemhat means "Amun is the foremost".

Sesostris I

nsw-bi.tj (xpr-kA-ra) sA ra (s-n-wsr.t)
The King of Upper and Lower Egypt (Kheperkare), the son of Re (Senuseret).
Kheperkare means "The Ka of Re has come into being". Senuseret (Greek version: Sesostris) means "The man of Useret". Useret is a reference to a goddess meaning "the strong one". It is often assumed that Useret refers to Hathor.

Sesostris III

nsw-bi.tj (xa-kA.w-ra) sA ra (s-n-wsr.t)
The King of Upper and Lower Egypt (Khakawre), the son of Re (Senuseret).
Khakawre means "The Kas of Re have appeared".

Amenemhat III

nsw-bi.tj (n-mAa.t-ra) sA ra (imn-m-HA.t)
The King of Upper and Lower Egypt (Nimaatre), the son of Re (Amenemhat).
Nimaatre means "The one who belongs to Maat is Re".

Seqenenre

nsw-bi.tj (sqnn-ra) sA ra (tA-aA qn)
The King of Upper and Lower Egypt (Seqenenre), the son of Re (Taa the Valiant).
Seqenenre means "The one whom Re has made valiant".

Ahmose

nsw-bi.tj (nb-pH.t-ra) sA ra (iaH-ms)
The King of Upper and Lower Egypt (Neb-peht-Re), the son of Re (Iahmes).
Neb-peht-Re means "Re is the master of Force". Iahmes means "the moon has born (him)".

Amenhotep I

nsw-bi.tj (Dsr-kA-ra) sA ra (imn-Htp)
The King of Upper and Lower Egypt (Djeserkare), the son of Re (Amenhotep).
Djeserkare means "the Ka of Re is holy". Amenhotep means "Amun is at peace".

Thutmosis I

nsw-bi.tj (aA-xpr-kA-ra) sA ra (DHwtj-ms)
The King of Upper and Lower Egypt (Aakheperkare), the son of Re (Thutmosis).
Aakheperkare means "The shape of Re is great". Thutmosis means "Thot has born (him)".

Thutmosis III

nsw-bi.tj (mn-xpr-ra) sA ra (DHwtj-ms HqA-wAs.t)
The King of Upper and Lower Egypt (Menkheperre), the son of Re (Thutmosis, ruler of Waset).
Menkheperre means "The shape of Re remains". Waset was the Ancient Egyptian name for Thebes.

Hatshepsut

nsw-bi.tj (mAa.t-kA-ra) sA ra (Xnm.t-imn HA.t-Sps.wt)
The King of Upper and Lower Egypt (Maatkare), the son of Re (Joined with Amun, Hatshepsut).
Maatkare means "Maat is the Ka of Re". Hatshepsut means "Foremost of the noble ladies".

Amenhotep III

nsw-bi.tj (nb-mAa.t-ra) sA ra (imn-Htp)
The King of Upper and Lower Egypt (Nebmaatre), the son of Re (Amenhotep).
Nebmaatre means "Re is the master of Maat".

Amenhotep IV / Akhenaten

nsw bi.tj (nfr-xpr.w-ra wa-n-ra) sA ra (imn-Htp nTr-HqA-wAs.t) sA ra (Ax-n-itn)
The King of Upper and Lower Egypt (Neferkheperure Waenre), the son of Re (Amenhotep, de god who rules in Waset). This nomen was later replaced by the following: the son of Re (Akhenaten).
Neferkheperure means "the shapes of Re are beautiful". Waenre means "The One of Re". Akhenaten means "Ray of the Sun".

Tutankhamun

nsw-bi.tj (nb-xpr.w-ra) sA ra (twt-anx-imn)
The King of Upper and Lower Egypt  (Nebkheperure), the son of Re (Tutankhamun).
Nebkheperure means "Re is the master of shapes". Note that three strokes can also be used to indicate a plural. Tutankhamun means "the living statue of Amun".

Seti I

nsw-bi.tj (mn-mAa.t-ra) sA ra (mrj-n-ptH stXj)
The King of Upper and Lower Egypt (Menmaatre), the son of Re (beloved of Ptah, Seti).
Menmaatre means "The Maat of Re remains". Seti means "The one of Seth).

Ramesses II

nsw-bi.tj (wsr-mAa.t-ra stp-n-ra) sA ra (mrj-imn ra-ms-sw)
The King of Upper and Lower Egypt (Usermaatre Setepenre), the son of Re (beloved of Amun, Ramesses).
Usermaatre means "The Maat of Re is Strong". Setepenre means "The chosen one of Re". Ramesses means "Re has born him". Note how in the second cartouches the signs representing Amun and Re are placed together, making this variant of Ramesses' name a wordplay on the name of Amun-Re.

Ramesses III

nsw-bi.tj (wsr-mAa.t-ra mrj-imn) sA ra (ra-ms-sw HqA-iwnw)
The King of Upper and Lower Egypt (Usermaatre, beloved of Amun), the son of Re (Ramesses, the ruler of Iunu).
Iunu was the Ancient Egyptian name of Heliopolis.

Ramesses IV

nsw-bi.tj (wsr-mAa.t-ra stp-n-imn) sA ra (mrj-imn ra-ms-s HqA-mAa.t)
The King of Upper and Lower Egypt (Usermaatre Setepenamun), the son of Re (beloved of Amun, Ramesses, ruler (in) Maat).
Setepenamun means "the chosen one of Amun".

Ramesses IX

nsw-bi.tj (nfr-kA-ra stp-n-ra) sA ra (ra-ms-s xa-wAs.t mrr-imn)
The King of Upper and Lower Egypt (Neferkare Setepenre), the son of Re (Ramesses, who appears (in) Waset, beloved of Amun).

Psametikos I

nsw-bi.tj (wAH-ib-ra) sA ra (psmTk)
The King of Upper and Lower Egypt (Wahibre), the son of Re (Psamtek).
Wahibre means "the heart of Re endures".

Neko II

nsw-bi.tj (whm-ib-ra) sA ra (nkAw)
The King of Upper and Lower Egypt (Wehemibre), the son of Re (Nekau).
Wehemibre means "the heart of Re is repeated".

Amasis

nsw-bi.tj (Xnm-ib-ra) sA ra (iaH-ms sA-nt)
The King of Upper and Lower Egypt (Khenemibre), the son of Re (Iahmes, son of Neith).
Khenemibre means "Joined (with) the heart of Re".

Nectanebo I

nsw-bi.tj (xpr-kA-ra) sA ra (nxt-nb=f)
The King of Upper and Lower Egypt (Kheperkare), the son of Re (Nekhetnebef).
Nekhetnebef (Greek version: Nectanebo) means "His master is victorious".