A Phaeron (male) or Phaerakh (female) is a Necron who rules an entire Necron Dynasty; after the death of the Triarch following the War in Heaven, they are the highest-ranking Necrons in the galaxy, barring the Silent King himself.
Because the Necrontyr were so short-lived, there used to be a fair amount of turnover in this position (and in all positions in Necrontyr society, for that matter), but after the pact with the C'tan and biotransference, the present set of Phaerons are effectively immortal and ever-living (in fact, they have a special rule called just that), so the only way to replace one is to kill him or conquer his dynasty. Either one is a difficult proposition, as the Phaerons were given bodies of the finest, most robust construction, and rule over endless phalanxes and legions of warriors. However, this does nothing to prevent all manner of civil wars and political shenanigans as various nutty robot zombie nobles attempt to seize power for themselves.
Phaerons were added to the Necron army in the Fifth Edition Codex, as a way to give the Necrons some character. Note the pun on "Pharaoh," the ruler of an ancient Egyptian dynasty, as if you needed more proof that the new Necrons are Tomb Kings IN SPACE.
- Imotekh the Stormlord, Phaeron of the Sautekh Dynasty, the most powerful Dynasty as of the 41st Millennium.
- The Silent King, the only surviving member of the Triarch and one-time ruler of the entire Necron Empire. After defeating the C'tan, he left the galaxy for parts unknown, met the Tyranids, and hustled back home to rouse the Necrons against the extragalactic threat.
- Trazyn the Infinite,
ThiefCollector Extraordinaire. Was never officially crowned Phaeron, but is more than powerful enough to deserve the title. (This is brought to you by Trazyn simps.)
Phaerons on the tabletop are basically Necron Overlords with an extra upgrade that makes them (and any unit they join) Relentless. In 5th this was an improvement, making your I2, 1-attack zombiebot horde suck slightly less when push came to shove. In 6th, assault was essentially worthless, so there's no real point in doing this. Unless you WANT to assault with warriors (pro-tip: don't bother).
In 7th Edition, you can't make an Overlord a Phaeron anymore. At all. Crunch-wise, it's no big deal (Relentless wasn't actually needed that much). Fluff-wise, you can just call your Overlord a Phaeron all the same.
EDIT: GW has heard our pleas, returning Phaerons to the tabletop as an upgrade as of ninth edition, this time in the form of a two CP upgrade that allows them to My Will Be Done twice in a single turn!