Gnome Stew has published a list of “The 12 RPGs every gamer should play before they die.” It’s decent, but there are some that duplicate experiences and some lamer ones. There’s also a storygames thread with other peoples’ takes.
My personal list. Remember, this isn’t Best Games (though some qualify as that too) – it’s the games that you should play to broaden your horizons about gaming:
1. D&D, some variety. It accounts for probably 90% of all roleplaying groups, and you miss out on the “RPG experience” if you haven’t played it.
2. GURPS, HERO, Silhouette, or similar high crunch “universal” game system, to see what that’s like and if it’s your thing. Here, you will learn the wonders (?) of heavy simulation.
3. Feng Shui, for a good “gateway to player narration” game for those who have played too much of the previous 2 choices and their “trad” ilk (Palladium, White Wolf, etc). Or maybe Savage Worlds, but I haven’t played it so can’t vouch for it personally. [Edit: No, I’ve played SW now and I like it fine and it’s rules-light and all, but it’s still firmly traditional.] Feng Shui opened my eyes to a whole different way of playing. “I’m not totally dependent on the GM for every detail of the world? And I’m already a badass so I don’t have to worry so much about survival or level grind and can focus on cool? You’re blowing my mind, man.”
4. Over the Edge – for my money, a better pick than fan favorite Amber for a fully freeform kind of game. Amber just has always been too “Zelazny-ey” for me. Admittedly, Amber is more purely democratic and negotiation-based. It’s not a bad choice. But I personally like OtE.
5. Paranoia or Toon or Ghostbusters or InSpectres or octaNe or Star Thugs – for stressing silliness and fun over all the rest of the nonsense. Sometimes you want to be seriously in-character and all deep, but sometimes you want to make fart jokes.
6. Call of Cthulhu – for generally the best horror experience in gaming and for a system that is both somewhat crunchy and can be explained to someone in 5 minutes. I prefer the classic BRP version in this context.
7. If you love a specific genre, then try some supers (Mutants & Masterminds), scifi (Traveller), modern (Unknown Armies), espionage (Top Secret, Spycraft) and/or western (Aces & Eights, Deadlands) game. Or whatever genre you’re into; pulp, cyberpunk, etc. all have a couple excellent games that offer them. At the extreme end here are games like Buffy, James Bond and Star Trek:TNG that show how to even more slavishly reproduce a specific literary/TV/etc genre. What you want to learn here is how a game’s rules and setting can be used for highly specific genre emulation.
8. One random hippie indie game, to get it out of your system 🙂 Spirit of the Century if you’re delicate, one of the more random ones (Grey Ranks, etc.) if not. Mainly you’re looking to shake yourself up; so many RPGs end up falling into the same general paradigms and it’s cool to play something very different so you understand what’s possible.
9. Lacuna, Don’t Rest Your Head, or some other game where even the GM doesn’t know what the hell is going on. Thought experiment play isn’t everyone’s cup of tea, but you should at least try it.
10. Warhammer Fantasy Roleplay, or one of the other rare super-gritty games (Twilight: 2000…), to clear your palate from games where you can actually accomplish something, and reorient you to your pathetic, menial role in the universe.
In my opinion, it’s these games that open your mind the most to what RPGs can be like. From rules heavy to almost no rules, from totally GM-driven to heavy player narration, from serious and gritty to high action to super silly. Once you’ve played one of each of these categories, you can claim to have knowledge of the length and breadth of RPGing!
You could argue that you should play a “bad” game too, just so you know they exist and can identify pitfalls more readily – but I think you won’t have to go too far out of your way for that. RPGs are 80% execution, so chance will ensure that one of the games above will be a sucky experience for you regardless of it being a fine set of rules in a pretty book.
Side note. What is all the love for Dread in everyone elses’ lists? It seems awful gimmicky to me. “Let’s replace any kind of mechanic that takes difficulty or character skill into account, and replace it with something completely arbitrary (a Jenga tower).” Isn’t rock-scissors-paper LARPing like this? Why is this innovative? It seems to me like a “amusing to play, once” kind of thing.
Geek Related 
This discussion would be incomplete without a mention of ‘Bunnies and Burrows’ somewhere. So now I’ve mentioned it, so there.
“‘Let’s replace any kind of mechanic that takes difficulty or character skill into account, and replace it with something completely arbitrary (a Jenga tower).'”
I can’t account for anyone’s taste in games, but I can clarify that Dread does not replace difficulty or character skill with a Jenga tower. It replaces dice with a Jenga tower. The difficulty of the task at hand and the character’s skill in accomplishing the task are both still taken into account in much the same way they would be for a die roll.
I decided to edit and update/expand this a bit today. Hi, all you recent foreign visitors who’ve been discussing this post in your moon-man language!