Bunnyhop
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Bunnyhopping (also known as bhopping) refers to a movement exploit found in several 
Quake Engine, 
GoldSrc and 
Source Engine games, however it is not exclusive to these engines.
Originally found in Quake, bunnyhopping allows the player to achieve speeds far higher than running regularly, and is used heavily by speedrunners to complete levels faster.
How it works
Bunnyhopping is performed by timing a jump as soon as possible when hitting the ground. Doing so preserves speed, and if done on the same tick as the player lands on the ground, will result in no speed loss, commonly called a "perfect jump".
Air-strafing is done by using a directional key while moving the mouse in the same direction (i.e. moving left while turning left), which builds speed while in the air.
Combining these allows the player to gain much more speed than is normally possible.
Binds
Because timing your jump to be on the same tick as you hit the ground is exceedingly difficult (for example, on a server with a tickrate of 66 you have a timing-window of 1/66th of a second,) it is common to bind jump to the scroll wheel in order to send many jump inputs quickly for a greater chance to hit a perfect jump.
In Source, open the Console and use
bind mwheelup +jump;bind mwheeldown +jump;bind space +jump
to bind jump to both mousewheel directions and the space bar.
Scripts
Scripts can be used to automate bhopping and will guarantee a perfect jump every time. This is usually considering cheating and can result in a ban, whether by server admins or by Valve Anti-Cheat, however there are various gamemodes and speedrun categories where use of these scripts is allowed and encouraged.
Attempts to stop bhopping
Over time, 
Valve has attempted to minimize or stop players from bunnyhopping, with mixed results.
In 
Half-Life 2 and 
Portal, the game will attempt to slow down the player if they exceed a certain speed. It does this by applying a backwards force to the player, but because of an oversight, the game applies this based on the direction the player is facing instead of the direction they are going. The player can abuse this by turning around and bhopping while facing backwards, applying the force and adding to their current velocity. This is referred to as Accelerated Back Hopping (or ABH for short).
Later games on the 
The Orange Box engine branch or newer would implement more fixes to make bhopping harder and less effective, though still not impossible.
Gamemodes
Several multiplayer Source games, such as 
Garry's Mod and 
Counter-Strike: Source, have some servers dedicated to a Bhop gamemode (Garry's Mod even having a category in it's server browser for it).
These servers will typically feature maps designed as courses for players to race through, with plugins to track time, speed, and other information. These servers will also usually change certain cvars to change how the player controls while bhopping, such as sv_airaccelerate and sv_maxvelocity.