Wikipedia defines statics as: the branch of mechanics concerned with the analysis of loads (force, torque/moment) on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity. When in static equilibrium, the system is either at rest, or its center of mass moves at constant velocity.
In short, statics is the study of rigid bodies that are stationary, compared to dynamics which is the study of bodies in motion. A stationary body will have no unbalanced forces acting on it. For example, while you're standing gravity is pressing your weight down to the earth and the earth is resisting that force, causing the system to be in equilibrium (this is a static problem). If you were to jump the force of your jump would be greater than gravity for a few seconds allowing you to rise into the air (this is a dynamic problem). Another example would be a parked car would be considered a static situation while a moving car would be considered a dynamic situation.
The main idea behind statics is that the object is stationary. For that to occur all of the forces on the object must be in equilibrium. Equilibrium occurs when the resultant force and moment vectors are zero.
Therefore a system is in equilibrium when: F_{R} = \sum F = 0
M_{R} = \sum M = 0

Or another way of putting it is:
where: