By definition, work equals to zero when the force acting on the object acts on it perpendicular to the direction of its movement. The easiest way to conceptualize this effect is with uniform circular motion, and more specifically, centripetal force. With centripetal force, the force is directed away from the centre of the motion while the object moves perpendicular to it in a circular arc.
Another situation in which work is zero despite a nonzero force is when the displacement of the object is zero. This is seen in the equation of work:
W = F * d = F * 0 = 0
Pushing against a wall or an object sitting on a surface are examples of this kind of zero work with a nonzero force. When you push on a wall you are exerting force; however, the wall does not move. When an object is sitting on a surface, the surface exerts a force upwards on the object; however, the object exerts an equal force in the opposite direction, and a result, the object does not move.