Role-playing games were the gaming fad of the 1970s and early 80s: One player (the "game master") crafts a scenario which other players engage by playing characters designed according to a rules system. While there are rules for handling combat and rewards (treasure), the heart of the game is intended to be role-acting the characters. Of course, in the hands of kids like myself (at the time), it often became a combat game with treasure as a reward. What we really wanted were better wargames, but we didn't know that at the time. It would be another 10+ years before Robo Rally cam along to satisfy our needs. I'm not much of a role-playing game fan these days. I'd rather spend my time reading (and occasionally writing) fiction than playing RPGs. But I used to play them a lot. Here are some of the ones I played in "the olden days".
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Based on the fiction of early-20th century horror author H.P. Lovecraft, Call of Cthulhu is my favorite RPG. The nice thing about this game is that questions of game balance that plague other RPGs are rendered moot because most of the threats the characters face are a hundred times stronger than them anyway. CoC goes for realism - guns kill, and quickly - and pulls it off very nicely.

What this leads to is a game where thought and problem-solving are key, rather than going in and blasting things. And, of course, being a science fictional horror game, it lends itself to a high degree of inventiveness, especially as it's less fettered to its source material than some other games. The character design system is very nice, too.

I think this is a good game.

Champions

Champions is the role-playing game of superheroic adventure. Its selling point is that players get to create their own characters, fully and without any die-rolling. That in itself is an awful lot of fun. Additionally, for people who are familiar with and fans of the better examples of the genre, the superhero world lends itself very nicely to role-playing. The combat system is fast and a lot of fun, and definitely should be played with miniatures.

I'm mainly familiar with the second edition, with its various "expansions". The big problem in Champions was always figuring out how to handle money, since the goodies that Bruce Wayne-level wealth could buy would seriously unbalance the game. But this always seemed a small problem (and Dragon Magazine published an interesting solution to this problem, back in issue #73 if I recall correctly). The later editions may be better.

I think this is a good game.

The current edition of this game seems to be Champions: New Millennium. The editions I played (second and third editions) are considerably older.

Dungeons & Dragons

Okay, it's the original role-playing game, played by thousands, if not millions, of people. But is it any good? Well, the fantasy milieu is not a problem (although it doesn't lend itself to actual role-playing as well as some, in my opinion), but the rules are severely burdened by having been the first, and being thus crammed with many ideas that didn't pan out. The implementation of armor and combat is clumsy, and spell system is contrived (albeit almost entirely lifted from Jack Vance's book The Dying Earth), and the experience system is so ridiculous as to defy belief.

I admit, I played D&D a lot when I was younger, and enjoyed it quite a bit, too. But compared to many games on the market, it's not particularly good, and it feels a bit too generic. (A friend tells me that the current Third Edition improves considerably on the Advanced D&D I played in my youth, though.)

I think this is an okay game.

One of the earliest SF RPGs, Gamma World takes place in a post-atomic holocaust world, with radioactive creatures, demolished buildings, remnants of civilization, and various mysteries to be uncovered. The game system is not much different from that of Dungeons & Dragons, and suffers from similar problems. Plus, the mutated creatures seem all too generic, and often not very imaginative. Although the rule book is fun to read, this game never struck me as terribly playable.

I think this is a poor game. I think it's long out of print.