The d20 System is a tabletop RPG system that uses, among other things, twenty-sided dice.
Most of the d20 System was released as the System Reference Document (SRD) under the Open Game License (OGL) as Open Game Content (OGC), which allows people to make and release supplements without having to get a license. This led to multiple game-breaking splatbooks for D&D and a glut of samey RPGs, mostly because people figured it was worthless to challenge Wizards, and so d20 was soon seen on everygoddamnthing.
Mechanically, d20 is basically a watered down, reversed version of the AD&D 2E rules mixed with mechanics (like feats) stolen from Megaversal and skills from Alternity. Watching it in motion, however, is enough to make a mathematician orgasm. In the D20 system, you can break everything into nice, easy-to-understand percentages. Because 20 is exactly 1/5 of 100, the outcomes of a D20 roll are complete percentage points when you're trying to work out the odds of something happening. Simply figure out your target number (example: A DC of 10) figure out the number of results that are higher than the number (10), add one since D20 system lets you succeed when you match the result (11), multiply the result by five, and you have your final unmodified percentage chance of success (in this case, 55%). Each +1 you have gives you an additional 5% chance of success to any roll - simple and elegant. By comparison, a d6 has a 16.66% boost per roll, and the D12 is +8.33%. And don't get me started on 2d6 or 3d6 curves... The most complex thing is figuring out how Advantage works, but aside from that it can all be done in one's head or on any phone. If one can accurately guess the rough DCs of any challenges that you might be up against you can play the odds easily and quickly in your head. Not all systems can say this. The fact that the math of the D20 is so easy benefits both the players who are fast enough to play the angles, and DMs who set up challenges so they know exactly the odds of the players succeeding.
...barring twenty 1's in a row, because Chaos theory and Lady Luck work like that some times.
Wizards doesn't use the d20 System any more, as they decided to
fail fuck everyone over change the mechanics for D&D 4th Edition. However, there's still hope for people who RAGE at 4e, since Paizo Games pretty much stole 3.5 and renamed it Pathfinder.
After their temporary bout of insanity, WotC came to their senses and released 5E, releasing roleplaying games to the mainstream - and going back to what is clearly a lot of 3.X-sensibilities, with a splash of 4E, and even AD&D thrown in for good measure.