Zerg Overview

I think that the Zerg look way cooler than the other races in StarCraft. The Zerg have a centralized center of unit production; no matter the unit, it is going to be built/grown from a Hatchery/Lair/Hive. This means, build multiple hatcheries as soon as possible, so that you can have plenty of larva ready to mutate. Also, since the Zerg 'grow' their troops, and for the most part they are cheap, they depend on mass numbers of them (especially in the case of zerglings) to win battles; don't expect to make a few zerglings and win.

The Creep
The Creep is pretty cool. Since all of the Zerg buildings are living creatures, they can only be placed (with the exception of a Hatchery) on the Creep, which provides nourishment for the buildings. The Creep can also warn troops that the Zerg are present; this can be helpful when scouting, as you can be out of range of Zerg troops, but still know where the base is.

Regeneration
One of the Zerg's most helpful qualities is the fact that all units and buildings, since biological in nature, regenerate over time. This means that a building that was damaged in an enemy attack will be at full strength in a relatively short amount of time. Also, units in battle that are damaged can be burrowed until they heal, then they can attack at full strength. When attacking the Zerg, make sure you kill EVERYTHING, since it is this healing factor that can make the Zerg near impossible to wipe out; a hatchery that has 1 hit point is still a threat; don't leave until everything is gone.

Control
The Zerg's capacity device is the Overlord. The Terrans build Supply Depots, the Protoss warp in Pylons, and the Zerg grow Overlords. The Overlords provide psionic control over Zerg troops; they are also detectors. Not only that, but they are a flying unit, meaning that you can start scouting as soon as you start. Also, once this is researched, you can carry other units in Overlords. Overall, don't refer to an Overlord as just your supply unit; consider it scout, carrier, and supply unit. But they are good and cheap, so don't worry if they get killed.

Burrowing
All Zerg ground units (except the Ultralisk) can burrow. This can be very helpful. Attacking Zerg can burrow when the going gets tough; a computer player won't bother looking for those units once they burrow. Burrowed damaged units also can heal, meaning that they will soon be ready to attack once more. Basically, burrowing can be your best friend if you use it right.