Meaning of "Electrostatics"? I'm learning electrostatics right now in college physics 2, and I'm struggling to understand what "electrostatics" exactly specifies. My professor told me that "when charges are at rest in our frame of reference, they exert electrostatic forces on each other ", but what does it mean that the charges are at rest in our frame of reference? Does that mean that point charges are not physically flying around in space? Like the "radius" value in couloumb's law between two charges is not changing? Or does charges being at rest mean that there is no 'flow' of charges like two metal balls w/ charge 2 couloumbs and -3 couloumbs being brought into contact and obtaining -0.5 couloumbs each? 
What happens to Coulomb's law and Gauss's law when charges are not at rest? 
Also, why is the electric field inside a conductor zero? This sort of branches out from the definition of "being at rest". My textbook says that "If there is an electric field inside a conductor, the field exerts a force on every charge in the conductor, giving the free charges a net motion. Since by definition electrostatic situation is one in which charges have no net motion, the electric field inside the conductor must be zero". What does that mean? If there is an electric field inside conductor the electrons will be flying out of the conductor?
 A: You see, there is a general theory of electrodynamics which tells you how moving charges and currents create electrodynamic fields, which is governed by Maxwell's equations and the Lorentz Force Law. These generalize Coulomb's law and so on in the case that everything is not static.
Now electrostatics is a special and much simpler case of this, and in many situations it's enough to study this. So be patient, you will learn the more general theory later, I hope, or you can look in a book like Griffiths, electrodynamics. In any case, it's good to study the simple case first since it is much easier to understand.
To your second question: No, the electrons would not fly out of the conductor if there was a field. They can move freely inside the conductor, so they will just move around until they reach a place where the net fields are zero again. But if you want to study electrostatics, the only static situation is that the field inside a conductor is zero - if it is not, the electrons must move around and not be static.
(Next time, please split up into two questions.)
A: "Electrostatics" is the study of electricity with charges at rest or "nearly" at rest.  By "nearly" at rest we understand that the velocities of the charges are much less than the speed of light.  
Moving charges produce magnetic fields but this magnetic field is usually neglected when $v/c\ll 1$.  Gauss' law (in vacuum)
$$
\oint \vec E\cdot d\vec S=\frac{q}{\epsilon_0}
$$
remains unchanged when charges are moving because the right hand side contains the net charge (the value of which is independent of the frame of reference) and $\epsilon_0$, which also has the same value in any frame.
