The equilibrium constant Kc=[Products]^coeffcient/[Reactants]^coefficients
In words: The equilibrium constant is the concentration of the products raised to the power of the coefficient in front of them of the balanced equation divided by the concentration of all the reactants raised to their coefficients. This is of course the "easy" explanation of an equilibrium consant, you probably don't want to see the P-chem explanation.
I have no idea how to use Latex, so it would almost be impossible to write down how it is derived. That website shows how to derive the equilibrium constant for gases which will give you an idea of how it is derived in general. You have to work with chemical potentials etc. to derive the equilibrium constant.
Your chemistry text should explain this in simple terms, the easiest way to understand equilibrium constant is in terms of the rate constants. The derivation from the forward and reverse rate constants should be given in your text. It is quite important that you understand the relationship between rate constants and equilibrium constatn.
It seems that they have explained the rest. although it is a bit more complex . You should study this section more in depth.
Chem tr. where did you get the redox example (your first post)?