What is difference between L2 norm and H2 Norm?

I am not sure about your application -- and we say the $L^2$ norm of a function and not a system. But for simplicity I will explain the concepts for real valued functions. Consider an open domain $Omega$ and a function $f:Omega to mathbb{R}$. We say that $f in L^2(Omega)$ if $||f||_{L^2(Omega)} < infty$ where begin{equation} ||f||^2_{L^2(Omega)} = int_{Omega}|f(x)|^2 dx. end{equation} Intuitively, an $L^2$ function is a function whose area under its graph is finite while you allow discontinuities in the function itself. For example if you take a sine wave and create a set of discontinuities of measure zero (delete "some" points on the sine wave), it will still be integrable and hence in $L^2$ (yet it is not continuous anymore).

Now if you consider the space $H^1(Omega)$, it consists of all functions $f$ such that $||f||_{H^1{Omega}} < infty$ where begin{equation} ||f||^2_{H^1(Omega)} = int_{Omega}|f(x)|^2 + |f'(x)|^2 dx. end{equation} Intuitively, functions in $H^1$ are functions that are weakly differentiable, that is they are differentiable everywhere except at a set of points of measure 0. That means that $f'$ has "some discontinuity points" and so $f' in L^2$. (a very nice example is the hat functions)

Finally, using the same logic, functions $f in H^2(Omega)$ are those functions that are twice - weakly differentiable and so the same logic of the previous space $H^1$ applies.