The percentile die is a part of the traditional 7-dice D&D dice set and several games use the percentile roll as the primary roll in the game, including Call of Cthulhu and Eclipse Phase -- though those games usually include other dice for things like damage rolls.
Twelve-sided dice are a part of the traditional 7-dice set and are used in a myriad of roleplaying games, though the d12 rarely takes a very prominent roll. 12 sided dice are used in RPGs like Dungeons & Dragons / Pathfinder, Earthdawn, and Call of Cthulhu (where it's likely only used for sanity loss rolls).
D20, twenty sided dice. These dice are the darling of the RPG industry, ever since the original Dungeons & Dragons was published in 1974. As the basis of D&D and also subsequent role playing games using the D20 system, the 20 sided dice has a commanding place in the world of gaming dice.
The d4 is a platonic solid, the tetrahedron. Gamers sometimes colloquially refer to 4 sided dice as caltdrops and most veteran gamers have accidentally stepped on the sharp top edge of these dice at least once. Non-gamers often call them pyramid dice.
The six-sided dice (also known as a d6) is manufactured in more colors, sizes, and styles than any other type of dice. This isn't surprising, considering that the humble d6 has been used for entertainment, gambling, and divination purposes since the dawn of recorded history.