In physics, equilibrium refers to the state of any object when all forces acting upon it result in zero change of motion for the object. Static equilibrium indicates that the object in question is motionless. Dynamic equilibrium indicates that the object is moving, and will continue to do so unchanged.
Static equilibrium is generally easier for people to understand. Consider a cup on the edge of a dining room table - it is in a state of static equilibrium. The forces of gravity that would cause it to fall down are perfectly counter-balanced by the force of the table holding it up. As a result, it moves in neither direction, and will remain in this state until something changes. For example, imagine that a dog bumps the table, offsetting the balance of the cup. It topples over, rolls off the table's edge and falls to the floor.
Dynamic equilibrium is a little trickier to conceptualize because examples are not as easy to imagine on earth. Imagine a jet flying through the sky when there is absolutely no air movement whatsoever (a condition highly unlikely ever to exist). In this example, the jet has four basic forces acting upon it: 1) gravity, trying to pull the jet down, 2) lift generated by the jet's wings, trying to pull it up, 3) thrust from the jet's engines, trying to propel the jet forward and 4) air resistance, trying to push the jet backwards until it stops its forward motion. As long as the jet flies perfectly level, there are no changes in the air around it, the engines produce a constant level of thrust and the force of gravity beneath it does not change, the jet will settle into a "dynamic equilibrium" where its up/down motion remains zero, and its forward motion will not change. It is moving, but the rate and direction of movement are constant.