First off, we have the bird people from Red Dead Redemption that just seems to be a part of the game. I've never played it, but I've seen videos and it makes me laugh. Basically the game has mapping problems, so it attempts to map people models onto animals, with naturally hilarious results.
Backwards long jumping is a method in Super Mario 64 that allows you to finish the game with only 16 stars, as you move backwards up sets of stairs so fast it jumps you through the game's star doors. It's hard to do, and for a while me and my mates used the phrase "backwards long-jumping" as parlance for impossible.
Missingno is a 'pokemon' that can be summoned in the original GB Pokemon games by using the HM Fly move to land on the shores of the Cinnibad and Surf around the coastline and with any luck the missingno will appear as a skyscraper like apparition of pixels and basically has the potential to really mess up your save file and has an odd side effect that if you encounter him it maxes out an item in your inventory to x99 amount so became commonly known as the Rare Candy Cheat. Missingno has since become a small part of the Pokemon fanverse but I have so far not heard of any official endorsements.
The "swordless link" glitch involves fighting Ganon at the end of Ocarina of Time, and saving and quitting after he knocks your sword out of your hand. I like this glitch cos it simply screws up your game irrevocably. You can't get it back, it's that simple.
But the Ben's favourite tour de force of the Glitch-o-verse follows...