We can calculate the electric field in various circumstances using the familiar Coulomb's law or the Gauss' law. Although these laws are universally applicable, calculating the Electric field for a particular situation might not be easy using these laws. So we have another useful method to calculate the field and related quantities.
In our new method, we note that the line integral of electric field around a closed loop is zero (remember, this is true only in a static condition, that is, when the charges aren't moving), or equivalently, the curl of E is zero. Now, it is a theorem from vector calculus that if the curl of a quantity is zero, then that quantity is definitely the gradient of some scalar quantity.
If you aren't familiar with this theorem, or want some better physical picture, here's the argument. The work done by the electrostatic field on an object doesn't depend on the path that the object takes in the region of the field, It depends only on the starting position and on where the object ultimately lands up, the direction and magnitude of the electric field, and the magnitude of the charge.
Say the charge on the object is 1 nC and say the object is displaced 5 metres in a direction parallel to the field (even if the object moves at angle, we can resolve the displacement components and need to consider only the component parallel to the field). The work done on it is independent of the path it takes, we just need to ensure that it has the same displacement via each path. Say we find that work done on the object by the field is 1mJ (We can find it out by measuring the change in its Kinetic Energy which is also 1mJ if no other force is doing work on the object).
Now we take another charge, this time 5nC, and find out the work done on it (for the same displacement, of course), we find that the work done is 5mJ. The work has become 5 times. (It will actually turn out to be so.) This independence of path shows that we can define a scalar quantity at every point in space, independent of any charge, and if you carry a charge from one point to another, you can calculate the work done on it just by knowing the values of that scalar quantity at the two points (and the value of the charge!). This scalar is called the electrostatic potential.
The new method that I referred to, for calculating the field in a circumstance, is finding out the potential first, (since being a scalar, its easy to find out) and then from the potential, finding out the electric field.
Principles of Electricity and Magnetism
Introduction to Electricity and Magnetism