Body separation glitch
From Halopedia, the Halo wiki
The body separation glitch is a model and animation error present in Halo 2 and Halo 3. The glitch allows players to separate a playable character's upper and lower body. It may be performed when playing as either the Arbiter or the Master Chief.
Look Sensitivity Method
The most basic way to perform this glitch is to simply look down at your character's feet, and spin around. If the Look Sensitivity setting has been set high enough, the character's torso will spin faster than their lower body, effectively allowing the character's body to twist in unnatural ways.
Object Bump Method
On the level Gravemind, when you are in the Mausoleum of the Arbiter, walk up to the Arbiter's armor sarcophagus, and try to walk inside through the front, where Thel 'Vadamee got his armor out from in the Arbiter cutscene. Look down while you are doing this, and the Chief's legs will be completely separated from his torso.
This method can also be used as the Arbiter in Uprising. Before encountering the final pack of Brutes guarding the first entrance into the Forerunner hallways, the player may notice a ledge in a slightly out-of-the-way location. By jumping onto this ledge, facing away from it, and looking down, the player can see the Arbiter's complete legs model, which appears pinched at its top, where the Arbiter's waist would be.
The glitch can be performed on Metropolis using a different method; the method also works on the Scarab seen in The Great Journey. When you encounter the Scarab, jump onto it, and then wait until it stops moving. After the Scarab stops, go onto one of its rear legs. Walk into the body of the Scarab and move back a little. Look down to see your legs moving without you.
Find a Scorpion, and drive it onto a sharp incline, such as the rails opposite the bases in Waterworks, or between an edge and the ground. You must tilt the tank so that its cannon is facing upward. Then, jump onto the body of the Scorpion. Walk under the cannon, crouching, until you can see a cross-section of the cannon. If done right, your legs will walk out from under you and keep moving to walk forward, but they will be stopped by the joint between the body and the cannon of the Scorpion. Another way to see this glitch on the Scorpion is to find one that is flat on the ground, and jump onto the cockpit. Look down and move so that you are looking through the cannon. Your legs should be sitting in front of you.
Halo 3 Cold Storage Method
On the Halo 3 multiplayer map Cold Storage, find the yellow tube with the Flood Tank Form inside it. Use the Forge turret glitch to enter the tube. Look down; sometimes you will be able to see the bottom half of yourself.
When Bungie modeled the characters in the game, they modeled and animated the upper and lower halves of each character separately, to make it easier to animate a character turning without moving their legs -- that is, to allow a player to turn their character's head without the character having to reorient their legs. The upper and lower halves of the bodies are synchronized such that if a character turns enough, their legs will reorient (as they would for a person in real life). The glitch, then, is the desynchronization of each half of a playable character's body.
The bump glitches result from the fact that the player model's torso is hidden, but their legs are still visible. Were the torso still visible, the player would see it instead of just the character's waist.