I'm afraid it's not going to be a one size fits all approach here :). There are hundreds, if not thousands, of RPGs, and each has their own particular mechanics. The d20 one's (D&D) are certainly one of the more common, but by no means the only one out there. I'll see if I can't illustrate a few different basic mechanics.

Percentile - Every is based off a 1-100 roll. Normally each player will have a set of skills or characteristics, each rated at X%. Roll under (or perhaps over) that skill, and you succeed. If you don't, you fail. This can be futher modified by enviromental circumstances, or players condition or gear, to shift the target number up or down. If you succeeded (or failed) by a certain number, then the result could be better or worse than expected. Games I know that use this would include Fantasy Flights 40k series (Dark Heresy, Rogue Trader and the forthcoming Deathwatch) and Call of Cthulhu.

d10s v Target number - This is commonly seen in White Wolfs systems. Each player has a certain number of dice based on stat, skill, gear and circumstance. Roll all of them, and any dice which is Target Number or higher is a success. Sometimes, a 10 will count as 2 successes, or a 1 will remove a success. The more successes you get, the better you do. Games that would use this would include Old and New World of Darkness, Exalted and Scion.

Shadowrun uses a similar mechanic, except it uses d6s instead of d10s.

d10, roll and keep v Target Number - Slightly different mechanic, same dice. The player has a certain number of d10s, based on stat and skill. They roll them all, keeping a number equal to the stat. These kept dice are counted up, and the result checked against a Target Number. For example, I might roll 6 dice, keep 3 of them (total roll of the 3 dice say 17) and compare that to my target. If I had a Target Number of 15, I would succeed. If it was 20, I would fail. Examples of this would be Legend of the Five Rings or Seventh Sea.

The d20 systems, or at least the ones I am familiar with, roll a d20, add modifiers from your skills, and compare that against a Target Number. Beat the Target and you pass, roll under it and you fail.

A rarer one is roll 2 dice, take the difference between them, add that to a skill, and compare to target number. I'm familiar with this mechanic in Qin and Feng Shui.

To cut a long story short, there are quite a lot of different ways of rolling dice, and they are going to vary from game to game. It's probably better to find a game whose background setting appeals to you, then ask those playing it for an explanation of the mechanics.