The Zerg are a race of creepy insects determined to assimilate all advanced life in the Galaxy. They were originally peaceful, but they were turned evil, then turned back peaceful shortly before joining The League.
The Zerg were originally discovered by the insane Xel'Naga on the planet Zerus. They were gifted with Purity of Essence, whatever that is, making them the counterparts of the Protoss. The Overmind (Who is in no way related to the Gravemind) oversaw the Zerg's peaceful evolution. However, the Overmind was possessed by the Dark Voice and forced to attack the Xel'Naga, driving them away from their Galaxy. They then embarked to the Milky Way, seeking the Protoss, all part of the Dark Voice's grand plan to bring about the apocalypse.
The Zerg then took part in a three-way war between them, the Protoss, and Renegade Humans (Humans not in the League). They captured some strange Ginger Freak and 'infested' her, turning her into a humanoid Zerg. This was all part of the Overmind's cunning plan to free the Zerg from the Dark Voice. Soon after, the Zerg Homeworld of Char was attacked by the Protoss, and the Overmind's Right-Tentacle Bug was killed. This created a brief mental link between his killer and the Overmind, who then discovered the location of Aiur and launched a full-scale assault. He then relocated to Aiur, which was a good idea considering he had just started an invasion, and got himself killed by the Protoss.
The Zerg remnants were divided between the Ex-Ginger and some freaky Worm, leading to the Brood War. After the War's conclusion, Gingey christened herself the Queen of Spades, and proceeded to attack everyone again. The Three-Way-War raged for many more years. However, during an attack on Char, a magical device demolished a lot of Zerg and De-infested Gingey, turning her back to a relatively normal being but keeping her hair strange.
It was then the Gravemind went to work with his manly charms. He convinced Gingey to ally with The League and so began a peaceful co-existence. The Overmind's cunning plan worked, with the Zerg breaking free of the Dark Voice's control and returning to their peaceful ways once again. They are now strong allies of the League, and unsurprisingly they are best buddies with the Flood.