Power ups or powerups are special powers, if you will, that aid the player in a race. Although, not all of them aid. There are powerups of every kind: good, bad, useless, essential... Their effects range from changing the car's speed, to increasing the pedestrians' head size, to making the world act like a pinball table.
Powerups can be found inside pickups that are scattered around the levels. Collecting one will trigger the powerup's effect. Very rarely the player will be granted one without a pickup; cheating and missions for example.
Some powerups last for a limited time, commonly 30 seconds, and others can be stored for future use. There are even others that last the whole race. During their activated period, they'll perform their tasks. When the time/ammo is up, the player looses the powerup, but is free to collect another pickup containing it. Collecting a pickup of a powerup that the player already has will increase its duration/ammo, if possible. Some powerups like the Recovery Vouchers in Carmageddon only allow 5 at max. Other games allow more.
User operable powerups, also referred to as items are fairly self-explanatory - they are a limited supply powerup which can be activated at any time the player chooses, simply by pressing a single key. They are the closest thing the Carmageddon series has to weapons, as they include napalm launchers, repulsificators and mines.
For a list of the powerups in each game, read their corresponding page.