Zergling Rushes used to be very common before the
Spawning Pool (the tech building required to produce Zerglings) was patched in Starcraft v1.08 and the cost of it was raised to 200 Minerals (previously it costed 150 Minerals like the
Terran Barracks and the
Protoss Gateway). A common strategy was to build three
Drones (bringing the supply used to 7) and then building a Spawning Pool as soon as 150 Minerals were gathered, then waiting until the Pool finishes and producing six Zerglings (two per egg) and finally, attacking. The Rush is still possible, but it isn't as powerful as it used to be.
The Zergling Rush is actually quite easy to fight off. Worker units can handle an attack force of six Zerglings when
micromanaged properly (this is especially true for Terrans, who have workers with 60 HP as opposed to the lower HP amounts of the Zerg and Protoss). A good rusher knows this, so you usually see the rusher attack buildings or whatever offensive unit the enemy might have. While doing this, the rusher sets the enemy's base as the rally point for more Zerglings to head to after birth.
The Zergling Rush has a lot of psychological impact.. I started playing as Zerg because I kept getting killed by it back in the day.