General question about von Neumann algebras

Question

I am just introduced to the topic of von Neumann algebras in a second-year graduate course on operator algebras. It is defined as a strongly-closed $^*$-subalgebra of $B(H)$, but I read that it could as well been defined as weakly-closed, and others. The problem is that I have $0$ feeling with these different topologies. I know some defintions (SOT is the locally convex topology generated by the point-separating family of seminorms $\{p_x : H → \mathbb{C}, u ↦ \lVert u(x) \rVert \mid x ∈ H \}$). I just really don't know what this means. That is, what I should think of when I read these sentences. I know this is a very vague question (probably requiring a long answer), but any input is greatly appreciated.