A typical adventuring party (Courtesy Rafeejewel)

Role-playing games are supposed to let you be anybody and do anything. Then a five-pound rulebook is dropped in your lap that lists races, classes, and sharply divided special abilities only assigned to particular combinations thereof. In other words, you might actually have more freedom in real life than in the game.

Generally speaking, a class is a path that bestows a set of abilities, usually increasing in a series of levels. The higher the character's class, the more powerful that character becomes in what he or she does best.

TVTropes defines groups these classes into eight categories: Tank, Healer, Nuker, DPSer, Mezzer, Petmaster, Trapper, and Jack.

The Tank takes damage.  His job is to protect the other classes from harm.  Strictly speaking, the Tank does not necessarily have to be good at fighting, just defending. This is usually, but not limited to, the Fighter in tabletop role-playing games. In modern games, he's known as a soldier or mercenary.

The Healer heals. If the Tank's job is to have a lot of hit points, the healer's job is to make sure he doesn't lose any. This role is traditionally the cleric in tabletop RPGs and, oddly, often a female in many console games. In modern games, it's always the field medic.

The Nuker explodes...hopefully other people. His job is to inflict harm from a distance. In fantasy games this is usually the wizard, but in modern games this can be the guy who carries the biggest gun.

The DPSer stands for "damage per second." Unlike the Tank, who just has to stand there and be punished for being in the wrong place at the right time, the DPSer takes the battle to the opposition.  He inflicts damage frequently, although not necessarily every second. This can be an archer, a blaster who gets in frequent shots (instead of his Nuker companion, who unloads his artillery in massive blasts), or a stabby rogue. In modern games, this can be everything from a guy who punches fast and hard to the psychopath wielding the flamethrower.

The Mezzer controls the opposition. It can be paralysis, it can be sleep, it can be draining an ability score...whatever the effect, the result is the same: the target is impaired.  Traditionally this is limited to casters with the ability to enchant the opposition: witches, wizards with access to charm spells, or bards who can dazzle their opponents.

The Petmaster has pets.  Pets that fight.  These pets can be anything, from henchmen to killer beasts to summoned demons to the living dead.  The point being they fight on behalf of the Petmaster.  Like Mezzer, this role is less well-defined in tabletop RPGs.  Rangers and druids have traditionally held this role.

The Trapper inflicts damage asynchronously. His strength lies in his ability to defend against a pursuing foe, laying traps behind him as he goes. This has never been tabletop RPGs' strong point, if only because the heroic nature of conflict doesn't lend itself well to tricking a monster into falling into a pit. Rogues usually fill this role.

Finally there's the Jack, short for "jack of all trades."  The rest of that phrase is "master of none," and that's the Jack's strength and weakness. In theory, it makes him a perfect role for the leader. Bards have always had this role in the past, and 4th edition Dungeons & Dragons made an effort to bring that role into the mainstream with the Warlord.