Floating vehicles/player glitch

"Floating vehicles/player glitch" actually refers to two wholly separate glitches present in Halo 3: The Covenant Invisible Floor through Campaign mode, and Temporarily-Floating Objects which is replicable through Forge mode.

The Covenant Invisible Floor
On The Covenant, fight your way to the Scarab Tank battle and destroy both Scarabs. After that, instead of following the Arbiter into the Citadel, take a vehicle (a Ghost or a Warthog will do) over the artificial cliff behind the Citadel. This can be a little difficult, due to a ridge at the bottom of the upward slope; however, you can get over the ridge by using one of the sloping ramps attached to the side of the Citadel.

As you go over the cliff, you'll find that instead of falling to your death, you and your vehicle remain suspended in mid-air. Additionally, if you did this with a vehicle with no passengers, you can get out of the vehicle and you will remain floating in the air while the vehicle falls; you'll find yourself in a falling pose, only able to move quite slowly. In addition, in co-op, you can get a companion to shoot the vehicle you're in, causing it to spin around.

This will also work with the two flying vehicles (Hornet and Banshee); just let them settle on the invisible floor over the artificial cliff and get out.

To get out of this status, either kill yourself by throwing a grenade against a wall or return to a previous checkpoint.

Explanation
The explanation is actually very simple: your character is simply positioned atop an invisible surface. The collision detection engine realizes this (which is why the surface stops you from falling), but other parts of the game don't (which is why you remain in a falling pose, and can only move very slowly).

Temporarily-Floating Objects
The Temporarily-Floating Objects glitch is a phenomenon that causes undisturbed objects, like Fusion Coils, Crates, or even the Master Chief himself, to remain in the air after the surfaces underneath them have been removed. It is most commonly used in Forge, as a means of making immovable Forge objects float.

Forge Method (Player)
Start a Forge session with at least two players, who shall be referred to as Person A and Person B.

Person A needs to stay in Player Mode, while Person B enters Edit Mode. Person B should place some solid object onto the ground—one that Person A can jump on top of. Person A should then jump on top of the object, and remain completely stationary from this point on.

Person B should, from here, delete the object that Person A was standing on. Person A will not fall; instead, they will remain floating in the air until they are disturbed—that is, until they are damaged, destroyed, shot, touched, or move of their own accord.

This also occasionally works by "Person A" standing on "Person B" (who should be in Player Mode), then shooting and killing "Person B".

Explanation
The glitch is the result of an optimization trick in the Halo 3 Engine. Like several physics engines, Halo 3's engine saves time on physics calculations by ignoring inactive objects. Specifically, if an object stops moving for a few seconds, the physics engine will stop paying attention to it. So long as that object is not disturbed -- that is, so long as it is not shot, touched, damaged, or destroyed, and so long as it does not move of its own accord -- it will not move. As briefly mentioned before, the glitch affects all movable objects, including Forge Objects, players, and even enemies -- when "time traveling" forward in Halo 3 's Campaign level The Covenant, players may spot Brutes that are floating in the air because they spawned in the air, and were not disturbed and did not move since their spawning.