What does it mean to say that "no work is done in moving a charge on an equipotential surface"? It is said that no work is done in moving a charge on an equipotential surface. It means that no force is required to make the charge move.
Then how could a charge be moved in an equipotential surface without applying any force? 
Evidently, the charge moves along the equipotential surface in the given case. But if the force isn't doing any work then how is the charge moving? Wouldn't it violate law of conservation of energy?
 A: The work that we are talking about is the one done by external force against the electrostatic interactions, or simply, the work by the electric field.
Of course, the external agent can do some work on the test charge along on the equipotential line, which would undoubtedly increase its kinetic energy. But no work will be done by the source of electric field when the test charge move on these lines.
There is no violation of the law here. 

Then how a charge could be moved in an equipotential surface without applying any force ?

To understand this, we have to bring a familiar concept into the picture.
Recall that in uniform circular motion, the centripetal acceleration is successful in changing the particle's direction of motion but fails to bring any change in it's kinetic energy (Because $\vec F_{cp}$ $\perp \vec{ds}$). 
Now all you have to do in this case is to make the external agent apply a variable force of appropriate magnitude and direction whose net resultant with the electrostatic force is perpendicular to the test charge's instantaneous velocity along the equipotential line. This process, however complex, will give the test charge the necessary change in direction of motion without any net work done on it. 
A: In addition to the mentions that the point of the statement is that no work is done by the electric field, you can also look at this with an analogy - on a frictionless horizontal surface I can move an object from point $A$ to point $B$ without doing any work by the time I'm done. The key insight is that energy I add to get the object moving I can later get back when I bring it to a stop, if I do it carefully. So, while I had done work during the process, that is canceled by a later negative work.
A: 
Then how a charge could be moved in an equipotential surface without applying any force ? 

If the force is everywhere perpendicular to the direction of motion then $\vec F\cdot d\vec r=0$ so no work is done.  Now, the electric field and thus the electric force is precisely everywhere perpendicular to an equipotential, hence $q \vec E\cdot d\vec r=0$.
A: 
But If the force isn't doing any work how is the charge moving

Motion doesn't require force. 
Acceleration requires force. 
If there is no force, ab object will continue moving at constant speed! 
Imagine a ball rolling on the floor. It doesn't stop. Nothing stops it so it just continues, until it hits the wall. That floor is an equipotential surface in terms of gravitational potential energy. The ball will not move upwards. That would require a force to do work so potential energy is gained. It also doesn't speed up. That would require a force to do work to add the kinetic energy gain.
In the plane floor itself it just continues rolling. Constant speed. No energy changes. No violation of the energy conservation law.
Think of a charge as that ball and think of the equipotential surface as that floor. 
A: Suppose a charged ball is gravitationally rotating on a circular orbit around a charged point source.
No work is done, not even the gravitational forces do any work. That is because the force points in direction of the point source and the direction of movement is tangential to the orbit, thus $\vec{F} \cdot d\vec{r} = 0$. Moreover, the eletric field points away from the point source, so no work is done by it either.
In conculsion, the ball is moving on an equipotential surface. Perhaps you haven't considered that the ball may already be in motion in the first place.
A: No work being done as the test charge moves about is the definition of equipotential surface.
Work has to be done, or is released, when a test charge is moved between different potentials.
