Static equilibrium occurs when the sum of all forces and torque (the tendency for an object to rotate from the force exerted on it) on an object are equal, and therefore an equilibrium, or cancelling out of competing forces happens.
This means that the object will not move, or it will be static.
A classic example of this in an everyday situation is when two people are suspended on a see-saw which is perfectly balanced.
Static equilibrium does not have a function as such. It is a naturally occurring state which comes about as a result of a balance of forces. A great explanation of static equilibrium, including relevant equations (if you want to get a bit more technical!) can be found in the following video: