- Roll dice less than equal to your ability score or skill rating to succeed;
- ... But above any opposing rolls or difficulty number, if any;
- Critical or "mega" success is a die roll that matches your ability score exactly;
- A fumble is the highest roll possible (00 on d100.)
I say "as it stands now" because I remember it being a little different when I read and played a little of Call of Cthulhu and Stormbringer. I distinctly remember that opposed rolls worked differently: you consulted a table that basically did some math for you, because your chance of success was 50% plus 5 times (your score - defender's score.) Also, I remember the critical/mega success being the lowest possible roll on the dice, not the highest; I think there was also a "special success" which was something like 10% of your score, or something like that. Not quite as elegant as what Zak describes.
I think a large part of the reason people didn't flock to Basic Roleplaying, aside from the lack of elegance mentioned above in the early versions of BRP, was the numeric range. d100-based skills in BRP appealed to engineering types, but seemed a little too precise for most people. Plus, before they adopted that Rule #2 about rolling as high as possible without going over, it was a roll-low system; people seem to like the idea of rolling high, and especially rolling high when your upper range is limitless. I think this is why a lot of people are attracted to the d20 dice+adds approach, despite the need to do arithmetic: you can have really big numbers, and that is psychologically appealing. That's probably why Rolemaster had as much pull as it did, when it should have just been a quickly-forgotten overly-complicated system that wasn't anywhere near as elegant as BRP: standard rolls could go as high as 150 to 200, as I recall.
Of course, there are ways of making D&D more like this elegant approach. As I've said before, the mechanics are the least important part of a roleplaying game, and it's not really that hard to substitute another conflict resolution mechanic into the D&D structure. I have some ideas I'll address later.