A Zerg Hive Cluster is the designated base of operations for the Zerg, and functions in a similar manner to the military bases used by the Terrans and Protoss. The biggest difference, however, is that the hive is a giant organism, with its structures acting as its organs.
A new Hive Cluster starts from a single Hatchery, the most important hive structure which spawns new Zerg and digests minerals and Vespene collected by Drones. A Hatchery also generates a carpet of biomatter called "Creep", which nourishes hive structures built upon them (note that all structures except for Hatcheries and Extractors can only be placed on Creep). The Creep can be extended by placing new Hatcheries or small structures called Creep Colonies, which in turn can be mutated into Sunken Colonies or Spore Colonies for ground defese and anti-air defense respectively. The various warrior strains the Hatchery can spawn require certain structures that produce the respective strain's genome. For example, a Spawning Pool provides the Zergling strain. Also, certain requisite structures allow the Hatchery to mutate in order to access the advanced tech tree (the Spawning Pool lets Hatcheries become Lairs and the Queen's Nest allows Lairs to become Hives).
"Control" is the factor that dictates how many Zerg units can be supported, much like the Supply and Psi factors used by the Terran's and Protoss respectively. Hatcheries/Lairs/Hives generate a very small amount of Control, a disadvantage that is compensated for by the Overlords, which generate control whilst avoiding combat by flying to safety.
As well as granting a warrior strain's genome, certain hive structures can provide "evolutions" (read: upgrades) to enhance their respective unit (with the notable exception of the Evolution Chamber and Spire, which upgrade ground and air units in general). For example, the Spawning Pool evolves Metabolic Boost and Adrenal Glands upgrades to increase the Zergling's running speed and attack rate, and the Hydralisk Den evolves Augmented Muscles and Grooved Spines to increase Hydralisk movement speed and attack range. Some abilities, like Burrow, the Hydralisk's Lurker Aspect and the Mutalisk's Guardian/Devourer Aspects require research (for most abilities) or a structure upgrade (only for the Guardian and Devourer Aspects).
A common tactic used by the Zerg against the Terrans (generally) involves planting parasitic Zerg organisms onto Terran structures, which entrench themselves within the host building and feed on the inhabitants. The most prominent example of this is the infestation of Terran Command Centers. When a Command Center has sustained enough severe damage, a Zerg Queen can enter the structure and deploy aforementioned parasite, which takes total control over the center. An Infested Command Center serves two roles, them being resource collection and producing Infested Terran Marines, which act as running bombs that use suicide attacks as powerful as a nuke. Infested Command Centers can still use Liftoff to relocate to a new position, which is useful for getting the building to safety after infestation and moving it to the outskirts of an enemy base to churn out a stream of Infested Terrans quickly as long as resources hold.