Prestige Class
Prestige classes are a set of player options from the Dungeons & Dragons 3e DMG and other supplements. They are character classes with a specific set of requirements that a typical first-level character will not qualify for. Thus, they are typically taken as a multi-class at a later time, and can be anywhere between three and thirteen levels big, most being around 10.
The original intent of prestige classes was something like creating a specialty profession for a particular DM's game, with a unique set of abilities not found in the core rules, yet relevant to the setting. In practice, they are really just another way for powergamers to spend all their time optimizing their character sheets so they can have as many kewl powurz as possible. Another downside is that classes with a lot of class features will have these features gimped because of their new features.
They do however serve one useful function: they make it easy to spot munchkins in Dungeons & Dragons 3e. All you have to do is read a player's character sheet. If his list of classes reads like a reject Yu-Gi-Oh monster ("I SUMMON DWARF MONK PSYCHIC WARRIOR!"), then he is a weeaboo munchkin who thinks Katanas are Underpowered in d20.
Famous, Infamous and otherwise Notable Prestige Classes
Eye of Gruumsh
A ten-level class for Orcs and Half-Orcs who worship Gruumsh. It makes you rage harder than a Barbarian, give some bonuses to (half) Orcs fighting at your side, grants you a neat AC bonus and lets you spit acid at the faces of your foes. Given its requirements (being aligned to Gruumsh, taking out your own eye) are not often what a player character does, they are more often seen as high-level enemies. The most notorious thing about them is that if an Eye of Gruumsh regains the ability to see with its eye it took it loses all of its class features, so if you face one cast Regenerate or other potent restorative magic on it to make it lose all of its abilities.
Grey Guard
A ten-level class meant for Paladins who do not like being Lawful Stupid. If you want to play your Paladin as Jack Bauer, breaking faces and chocking bitches to protect the innocent, this is where you should be at. Rare amongst prestige classes is that some of the features you get with the Grey Guard stack with your normal Paladin classes, making you suck less at your main job than you would normally. Features include being able to receive the Atonement spell without XP cost, allowing you to fall and get up again like a Necron in a Tubthumping video; being able to use Lay On Hands to hurt people (Lay On FISTS), smiting Chaos and eventually Good and Law as well, with the final bonus being unable to fall as long as you act for the forces of good.
Initiate of the Sevenfold Veil
A seven-level class meant for Wizards who want to become even more crazy powerful. While the class has notable requirements (12 ranks on two skills, meaning you have to be around level 9 before you can take this prestige class). It does not interfere with how the caster gains its levels, and makes them even more powerful. An Initiate gets to cast Wardings: shields they use as personal shields, area shields or walls. These wards can be infused with veils, much like the dreaded Prismatic spells: from fire, acid and lightning damage to constitution damage, petrification, insanity and disintegration. At higher levels the Initate can use these abilities more often, as reaction to being charged, have her Abjurations become more difficult to dispel, impart two veils on a warding and even cast Greater Dispel Magic infused with the power of the veils once per day, becoming more powerful the more magic effects the target has on it. Work together with your Cleric or Bard to make this more potent.
Mystic Theurge
A ten-level class designed to combine arcane and divine casters into a single package. Infamous for being a trap despite sounding good on paper: the Mystic Theurge requires you to take levels in both an Arcane (Wizard, Sorcerer, Bard and so on) and a Divine (Cleric, Druid, Paladin and so on), then the Mystic Theurge allows you to keep gaining levels in BOTH classes. The downside to this is you require to be able to cast level 2 Arcane and Divine spells alongside 6 ranks in Arcana and Religion knowledge spells, putting you at level 6 while only being able to cast as a level 3 class. This means that you'll always lag behind when it comes to casting levels, AND you do not get the delicious class bonuses/feats you normally would receive for leveling up. This puts you at a supreme disadvantage when compared to other classes: if you really want to play a class like just do some multiclassing or roll a gestalt character.
Rainbow Servant
A ten-level class that can give an Arcane caster access to the Cleric spell list. Based on the fake Native American myth about the rainbow warrior, who will defend all life (which was written by an evangelist, and said rainbow warrior is actually Jesus), the Rainbow Servant fights evil and inspire hope and mercy wherever they go. Like a paladin but not an ass and dressed in bright colors. In-universe the Rainbow Servants are the agents of the Couatl, Lawful Good feathered serpents who fight evil and despair all over the world. By finding their temples in the jungle they will train Arcane casters in the ways of Divine casting.
This class is, using the FAQ, more powerful than intended: the text describing the class says the Rainbow Servant gets new casting levels at EVERY level, netting you the full 10/10 instead of only 6/10. This, combined with the FAQ saying that text ALWAYS takes presedence over tables, means that you get the full package. Fun times! So at the cost of only two feats (as a Wizard) you get access to the Good domain and Detect Evil at will at level 1, the Air domain and the ability to grow multicolored wings at level 4, the Law domain and Detect Chaost at level 7, capping off at level 10 with Detect Thoughts(!) and access to all spells on the core Cleric list(!!). While this does not give you access to the Domain spells, this makes any Arcane caster FAR more flexible with what it can cast.
The use for the Prestige Class is somewhat limited for Bards and Sorcerers (since you can only CHOOSE the spells when you level, you do not get them for free), the class is rather good for the Wizard. You get a large number of spells to add to the Spellbook, making you a poor man's Archivist. Bring plenty of gold to write all those spells down when you hit the coveted level 10, though. But where the class truly shines is on the Warmage, Beguiler and Dread Necromancer. Since these classes get to pick their spells from fixed lists, this means that they get the full list of Cleric spells added to what they can pick from, immensely increasing their versatility.
Reaping Mauler
A five-level class that improves your grappling capabilities. It also allows you to cast a Sleep-like effect with your bare hands at level 3 and have a chance at outright killing a target by pinning it for three turns at level 5. The downside to this is that the saves for those effects are rather easy to make (Fort on DC10 + Reaper Mauler level + wis modifier). This means that for a good Reaping Mauler you'll need STR, DEX and WIS, meaning that you'll have to be a MAD Fighter, a Cleric or a Monk in order to use this class well. Still, with a bit of minmaxing you might end up pinning dragons to death.
Risen Martyr
A ten-level class from the Book of Exalted Deeds that turns you into Jesus. Well, more or less. You have to martyr yourself and, if you have the requirements (spread over saves, skills and feats) you rise from the dead as a Deathless, aka an undead who does not run on negative energy. They get all kinds of personal defense bonsuses and can cast shields. This goes on until level 10: if you gain enough XP to level up after that point you'll leave the world and go to the Upper Planes. Oh, and if you commit as much as 1 act of Evil (Exalted guidelines), your ass is grass and you get pulled to the Upper Planes for a paddlin'.
Ur-Priest
A ten-level class for Evil characters who want to steal Divine spells from the gods and use them against them. Being some kind of anti-cleric the Ur-Priest has a nice array of spells it can use (Cleric list) and cast (interestingly enough they're not bound by alignment). They also get to Rebuke Undead, get 20 Spell Resistance against Divine spells and spell-like abilities from Outsiders, can combine spell slots into higher level spell slots and at level 10 steal spell-like (no supernatural) abilities from any creature. While it is open to any class that meets it requirements, without some serious weight in Knowledge skills (and Bluff and Spell craft) you're not going to join the cool kids club.