The pharaohs are known to us today under their personal names, such as
Rameses and
Amenhotep, and there may be more than one of each. The
regnal numbers as in Amenhotep IV are modern and were not used by Egyptians, who did use an immensely complicated variety of royal titles and
epithets. Three of the most common they used were the
nsw-bìty or personal name, the
s3-r` or reign name, and the
h.r or
Horus name.
Essential note on language. Vowels were not written, so Egyptologists conventionally make words pronounceable by turning w into u, y into i, 3 and ` into a, and adding e elsewhere: so those consonant skeletons are said nesu-biti, sa-ra, her. In some cases the vowels can be deduced from Greek transcriptions, so you get variation. Where the conventional rule would give Ra and hetep and Imen, you often or usually see Re and hotep and Amen, Amon, or Amun. So there is no one right way of giving a name in English.
Two of the three common names were written in cartouches, resembling a speech bubble. This is how hieroglyphics were originally deciphered: the Rosetta Stone has parallel inscriptions in Egyptian and Greek, and the cartouches contain the names of Cleopatra and Ptolemy, which have a number of consonants in common to break the code. The Nesu-biti and the Sa-ra were written in cartouches.
As Gone Jackal says above, the Nesu-biti is conventionally translated King of Upper and Lower Egypt. It contains signs for sedge and bee, and precedes the cartouche. Formerly we used the Greek forms of the names, such as Cheops, Amenophis, Thutmosis, and Rameses. It is increasingly common to use a more accurate Egyptian form, Khufu, Amenhotep, Djehutmose, Ramesse. This was the king's birth name.
The name given to them on accession was the Sa-ra, "Son of Ra", and the signs s3-r` (a duck and the sun) were written before the cartouche. The Sa-ra name always ended in the god's name Ra, but because he was a god, his sign was written at the beginning. This is also true for all royal names, if they contain any god's name like Ptah or Amen. So Nesu-biti Tutankhamen had the Sa-ra name Nebkheperura, but this was written Ra-neb-kheper-w. This is called honorific transposition.
Instead of regnal numbers we can combine these two names and refer to e.g. Thutmosis III as Aakheperenre Djehutmose.
The common element kheper in many of these names is the scarab beetle, which is also used an abstract word of similar sound, meaning "become" or "form". Djehuty (picture of an ibis) is the god Thoth (which is the Greek form of his name). Ka (two hands upraised) is a spirit/soul, ankh (the familiar looped sandal-tie) is "life", neb is "lord" (and also means "every"), and tut is "image".
As well as the Horus name, preceded by the falcon h.r, there were two other lesser-used names, the so-called Golden Horus and Two Ladies names. Many other epithets were also used, extolling how beloved they were of this or that god, and how perfect and long-lived they were.