Adrenalin
There are a range of special moves which can be performed by your character under certain circumstances. Some require that you have 100 Adrenalin. Others, like the Dodging moves require that you have the Dodge setting ticked (See In-game Settings section). You can get Adrenalin points through various methods:

Adrenalin pill: 3 points
A Kill: 2 points
Headshots: 5 points
First Blood: 12 points
Killing sprees: 12 points
Once your Adrenalin reaches 100 points, you will be allowed to perform one of the following special moves, the effects of which will only last for a while. The following key combinations need to be pressed in rapid succession to activate the Adrenalin effect:

Speed: Forward, Forward, Forward, Forward
Regeneration: Back, Back, Back, Back
Invisibility: Right, Right, Left, Left
Berserk: Forward, Forward, Back, Back

Special Moves

Special moves require that you activate the dodging in the settings. Most of these moves need a lot of practice to be able to perform them at any time.

Double Jump: Jump and then press jump again in the arc of the jump.
Dodge: Tap Left, Right, Forward or Back twice quickly.
Dodge Jump: Perform a Dodge, and jump just before you lose your air.
Wall Jump: Run alongside a wall, Dodge (double tap) into it and as you're about to hit it, quickly Dodge away from it.
Goo Jump: Shoot some goo from the Biorifle, get a shield up using the Shield Gun, jump just when goo explodes (You can also use the shield gun to bounce energy weapons, break your fall and of course block normal weapons.)
Lift Jump: Jump while on a lift to get an extra boost.
Death from Above: If you jump from a great height and land on someone you'll cause them damage.