Bunny hopping is a technique that takes advantage of two things: Half-Life's air acceleration and a small "glitch" where if the player jumps at the same moment they touch the ground the game doesn't register that they have ever touch the ground at all.
Combining the two the player can then move at incredible speeds while only jumping, strafing, and a small motion of the mouse.
In order to do the bunny hop, though, you are going to have to have the original Half-Life disc as the bunny hop has been coded out of the games over the years through update. Mods exist that allow the bunny hop again but I have no experience with them.
Really the bunny hop is this: jump, hold strafe, move mouse along with strafe (so that you are moving forward/left), then jump strafe in opposite direction (so now you are strafing forward right as well as moving the mouse to the right.) Rinse and repeat. Don't hold foward or back; the entire forward momentum is created by your strafing motion. You're literally flying through the air as far as the game is concerned. Takes practice but once you get the hang of it you should be able to attain some good speed and be able to take corners and leap some pretty big gaps.
I think that works in Team Fortress Classic, too, doesn't it? And I'm pretty sure there are some HL mods that actually incorporate that on purpose. --- Please, call me Kelp. The_Mighty is just my title, not my actual name.
Bunny Hop only works with old versions of the Half-Life games (all Half-Life engine games) and with modern mods that allow the BH (often advertised as such.) The mods also allow bunny hop on source games, although the same effect can be achieved with about 750 gravity.