The definition of Electrostatics is the following:
Static electricity is an imbalance of electric charges within or on the surface of a material. The charge remains until it is able to move away by means of an electric current or electrical discharge. Static electricity is named in contrast with current electricity, which flows through wires or other conductors and transmits energy.
Electrostatics was at the inception of our knowledge of Electricity.
So, what is Electrostatics?
I have always been a Richard Feynman fan: The Feynman Lectures on Physics - Electrostatics, a lot of good information exists in the texts! A lot of our important Laws have come to pass, including Coulombs Law, Amperes Law and so on.
Onepower has kindly shared some good resources also:
Richard Feynman starts his physics lectures on Electrostatics of stating: [Topic: Maxwell's Equations]
The situations that are described by these equations can be very complicated. We will consider first relatively simple situations, and learn how to handle them before we take up more complicated ones. The easiest circumstance to treat is one in which nothing depends on the time—called the static case. All charges are permanently fixed in space, or if they do move, they move as a steady flow in a circuit (so ρ and j are constant in time). In these circumstances, all of the terms in the Maxwell equations which are time derivatives of the field are zero.
We must at this stage, review the definition of the Point Charge:
a point in space that has an electrical charge that exists only in theory and that cannot be measured
The Point Charge is crutial in Coulombs Law:
Coulomb's law states that: The magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them.
An important Law, Coulombs Law, uses a Point Charge that by definition can not be measured, a theoretical Charge.
Already we see, problems with Electrostatics! It is a field of Science that because of our need to simplify the Dynamic, we turned to Static, however in doing so, we lost part of the Science and ended up with problems.
In Electrostatics, because we have no motion, the Point Charge is stationary in Space, thus no Current, we have no Magnetic Field, and Science tells us that Curl vanishes! Another problem we see, not because of our lack of observation, but because of our decision to eliminate some fundamental concepts.
So how is it that Coulombs Law is important when we are missing so much? Richard Feynman on Seeking new Laws:
Again, what is Electrostatics?
It is Voltage, the Potential Difference of Charge, not Current, it is specifically Voltage.
The requirement for Voltage, is Charge Separation, Positive and Negative Charges must always be Separated. The same is true all through Science, and is not bound to Electrostatics by any means!
As Marathonman pointed out, I am still learning, and always will, for Nature is my teacher and Nature is very much smarter than I could ever be!