Widely used systems. Ok, here's the basics of dice rolling. There are three systems I'm fairly familiar with which have different mechanics for determining success.
First, and easiest, the percentile system. You have a stat or skill rated at a certain % value. Roll under that number, you succeed. Roll over it, you fail. Now, that number may be modified by outside factors, but you're still aiming to roll under it.
Example: I have a shooting skill of 40%. Any time I try to shoot someone, that's the number I'm working from. Now, maybe I'm at close range (+10%), but its foggy (-5%). My final roll is 45%. If I can roll 45 or less using a d100 (or more commonly 2 d10s) I hit my target. If I roll 46 or higher, I miss.
Second, the d20. The d20 system works by assigning a target number, which you're trying to beat. Your d20 roll beats that number, you succeed. If it don't, you fail. Again, the roll will be modified by outside factors ... skills, equipment, bonuses from high stats and so forth.
Example: I'm trying to bluff my way into somewhere I have no right to be. The GM decides the target number will be 20. Now, I have points in my bluff skill (+5), and a high charisma (+2). I now need to roll 13 or more on the d20 to bluff my way in. I roll a 10. Good, but not good enough, and I don't get in.
Thirdly, the 'X or better is a success'. In the White Wolf games, a 7 or better on a single d10 is a success. A 10 counts for 2 successes. I roll a number of dice equal to my stat + skill, count the 7s or better. I need to get X number of successes to succeed in whatever I'm trying.
Example: I'm trying to dodge out of the way of an attack. The attacker has gotten 4 successes. I roll my Dex (stat - 4 dice) and my Dodge (skill - 3 dice). I get 10, 8, 7, 7, 5, 3, 3, 2 . This gives me 5 successes, so I dodge the attack.
Now, frequently the better you roll, the GM will make the outcome more favourable to you. On the flip side, roll really badly, and it's likely to get worse.
How is that?