What is G and what is H? What is the difference in the way they are defined?

Both \(G(s)\,\) and \(H(s)\,\) are notations for transfer functions. Sometimes we use \(G(i\omega)\,\) (or \(H(i\omega)\,\)) to refer to the "frequency response" if the system is asymptotically stable; it can be obtained by evaluating \(G(s)\,\) at \(s=i\omega\,\).

--Shuo