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