A sequence of $L^2$ functions ${\psi }_k$ converges to $\psi$ weakly in $L^2$ if
$$ \int _{\mathbf {R} ^{n}}{\bar {\psi }}_{k}f\,\mathrm {d} \mu \to \int _{\mathbf {R} ^{n}}{\bar {\psi }}f\,\mathrm {d} \mu $$
for all functions $f∈L^2$.
What does it mean for ${\psi }_k$ to converge to $\psi$ weakly* in $L^2$ ? Same question regarding $L^\infty$.
Weak and weak* convergence are equivalent in reflexive Banach spaces, so for $L^2$ they are exactly the same. Since $L^\infty$ is not reflexive (in general), they are different in that case. Strictly speaking, weak* convergence is only well-defined if you say what space your space is a dual of. Typically, $L^\infty$ is viewed as the dual of $L^1$ (which it is by Riesz representation), so $\psi_k \to \psi$ weak* in $L^\infty$ means that $\int \psi_k f \, d\mu \to \int \psi f \, d\mu$ for all $f \in L^1$. On the other hand, $\psi_k \to \psi$ weakly in $L^\infty$ iff $L(\psi_k) \to L(\psi)$ for all continuous linear functionals $L$ in $L^\infty$. This space contains $L^1$ as a proper subspace, but it is in general impossible to describe explicitly, so weak convergence is a stronger and slightly hard to deal with notion of convergence.