d20 is the system devised by Wizards of the Coast for 3rd Edition D&D, where the success of challenging action is decided by rolling a d20, appling modifiers, and checking if it reaches the DC of the action.
As it's a very good system, many other roleplaying games have been based on it; in fact, Wizards of the Coast have released the core mechanics under the Open Game License, allowing anyone to use the d20 system. That's why you see lots of games like d20 Modern and Star Wars carrying the d20 system logo.
AD&D stands for Advanced Dungeons and Dragons. It's a refinement and reorganisation of the 1st edition Dungeons and Dragons rules which were at the time very disorganised as they consisted of a load of different systems bolted onto each other (the mechanics for actions such as lock-picking, sneaking, combat and crafting were all completely different and were written down in different books). The 2nd edition of D&D was also refered to as Advanced Dungeons and Dragons, and it's usually what people mean when they refer to AD&D, and what you'll see refered to on games like Baldur's Gate.
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