RPG
RPGs are traditional games, usually played with pens, paper and dice.
Rollplayan vs Roleplayan
There has always been some controversy whether RPG means a rollplayan game or a roleplayan game. Both categorisations are muddy, so what follows is a general overview of the two and should never be taken as final or completly accurate.
The advocates of rollplayand insist that RPG are, all in all, games and thusly throwing dice and numbers around is their central point of existence. Because nobody really finds moving numbers around and doing calculus fun in and of itself, the crunching usualy serves a purpose. Rollplayers are thus usually equated with hackanslashan, munchinism and optimisation. All of these views focus on the game's mechanic and creation of characters that use them to their best advantages or even exploit them. Why this is very unpopular with some people is because it operates under the assumption that the game must be somehow "won", while that is precisely what largely separates RPGs from most other tabletop games. Rollplayers tend to regard the GM as a source of challenges or even an enemy they must defeat by creating the most efficient characters possible. A less extreme view just insists that rolling dice and crunching numbers is simply fun.
Roleplayers tend to put more emphasis on the narrative, and insist that the dice stay in the backstage, operating from behind the veil with minimal or no intrusion in the actual "story" being played. This is why they are often called dramafags. A popular distinction puts rollplayers in the D&D campus while roleplayers are supposed to be WoD players, goths, wannabe "deep" etc. Roleplayers are also often ridiculed for writing extensive backgrounds for their characters "no one ever reads" and for falling to deep into the whole 'play pretend' aspect of RPGs. This also puts them closer to LARPfags whom nobody likes (except scandinavian ones, that shit puts hair on your chest). As with the previous example, this view consists mostly of stereotypes and generalisations.
The truth is, as always somewhere in the middle and there are always exceptions to the rules. Hardcore examplars of both sides can often ruin the fun for the rest of the people at the table either by propagation of their mary-sue fantasies nobody is interested in or by creating characters that beat everything up before anyone else gets the chance to act. Neither side is able to see eye-to-eye with the other because they operate under different mindsets and arguments will not strike the right chords as they're hearing different frequencies.
=GNS
A simmilar distribution has been proposed by the GNS (Gamist, Narrativist, Simulationist) theory which is not applied to gamers but the games themselves. GNS as a whole has been recently dismissed as bullshit by pretty much everyone; its individual categories can still be usefull for explaining some concepts however. Gamist RPGS focus on the "game" aspect, they are closer to wargaming and boardgaming in the sense that the rules or mechanics itself take the spotlight. Like you don't play Monopoly to pretend that you're a capitalist shit, gamist RPGs don't care about "getting in character" or playing in a "believable world" and so on. Gamism is best suited for beer and pretzels games and one-shots such as Kobolds ate my Baby. Narrativist games try to be quite the opposite. The storytelling and narration is the focus of the game and the mechanics are mostly or completely subdued to it. In Narrativism, you don't use a mechanical effect to perform an action, your action and its success tries to grow directly from your narration, while the rules just support this. Simulationist games try to "simulate" the workings and mechanics of a believable world. Simulationism usually takes an emphasis on realism, but since most RPGs take place in fantasy or sci-fi worlds, "believable" is the more correct expression. Simulationism takes pain in attempting to resolve actions and events in a way that would be expected "in real life"; this is usually accomplished by numerous tables and random rolls.
In reality most games feature a inseparable mixture of all these elements.
Various RPGs
The most well known RPGs around are D&D, GURPS and the Storyteller System, also known as World of Darkness. Other notable examples include Unknown Armies, Warhammer Fantasy Roleplay, Call of Cthulhu, Exalted, Traveller, Legend of the Five Rings, Burning Wheel, Riddle of Steel, 7th Sea, Spirit of the Century, Dogs in the Vineyard, Shadowrun, Savage Worlds, Deadlands, Mutants and Masterminds, Risus, Kult, Rifts, Donjon, My life with Master, Basic Roleplaying, Dying Earth, Little Fears, Pendragon, Amber Diceless, All Flesh Must Be Eaten, Feng Shui, Reign, Castle Falkenstein, Nobilis, Ars Magica and the infamous FATAL.