Electric potential and work 
A unit positive charge $q$ is placed in an electric field caused due to a positive charge $Q$.  $q$ experiences a force of repulsion $F_R$. We apply an external force ($F_{\rm ext}$) opposite to $F_R$. The test charge moves in the direction opposite to the electric field. Work is done opposite to the field. This work done is stored as potential difference.

Now my doubt is  below:
We know that $F_R=-F_\mathrm{ext}$, so two equal and opposite forces are acting on the charge $q$. If they are equal and opposite, they should cancel each other out now, don't they? Then how is work done against the field? Why is the charge $q$ moving in the direction of external force if the repulsive force (equal in magnitude) is opposing it?
 A: I don't understand why textbooks bring external forces into their treatment of electric potential. If a test charge $q$ goes from A to B in an electric field, the work done on it by the field is
$$W_{A\to B}=\int_{\mathbf{r}_A} ^ {\mathbf{r}_B} q \mathbf E.d \mathbf r$$
This is the loss in potential energy of $q$, because it is work that $q$, having gone from A to B, can no longer have done on it by the field. So
$$qV_B-qV_A=-\int_{\mathbf{r}_A} ^ {\mathbf{r}_B} q \mathbf E.d \mathbf r$$
[This is perfectly valid even if there are no other forces on $q$; in this case $q$ (or the particle carrying it) acquires kinetic energy equal to the electrical potential energy lost.]
A: Potential energy is associated with a system, not a particle.  One must calculate potential energy for pairs of interacting particles, not individual particles.
$F_{ext}$ is a force external to the system.  It can do external work and raise the energy of the system.  $F_R$ is work internal to the system (and a conservative one), and does not increase the energy of the system.
But there is a connection which can make the ideas  confusing.  The definition of potential energy is $\Delta PE = -W_\mathrm{int}$ when the work is performed by conservative forces.    But sometimes, as in your example, as a consequence of Newton's Third Law the internal work can be calculated by examining the external forces.
