What is the meaning of $L^{2}_{loc}(\mathbb{R}, L^{2}(\Omega))$?,
where $\Omega\subset\mathbb{R}^{d}, d\geq 1; L^{2}(\Omega)\equiv \mathcal{L}^{2}(\Omega)/ker(\|\cdot\|_{2})$
Is there some good reference (website could be better) to know the meaning of relatively complex symbols (generally used) like the above one?
I've found this definition (of local integrability):
$L^{2}_{loc}(\Omega):=\{f:\Omega\rightarrow \mathbb{R}$ measurable/ $f|_{K}\in L^{2}(K), \forall K\subset \Omega, K compact$}
References: https://en.wikipedia.org/wiki/Locally_integrable_function https://www.macs.hw.ac.uk/~simonm/funcnotes.pdf
But even with that definition, I cannot find a sense to the symbol $L^{2}(\Omega)$ in $L^{2}_{loc}(\mathbb{R}, L^{2}(\Omega))$.
These are very common function spaces calles Lebesgue spaces. In general, $L^2(X,Y)$ is the space of functions from $X$ to $Y$ such that their square is integrable (in the norm of $Y$ of course). As you note, functions have to be equal only almost everywhere to be equal in these spaces. $L^2_{\text{loc}}(X,Y)$ indicates that the integrals only have to be finite on every compact set (So it is not a global property. If $Y=\mathbb{R}$ or if it is clear from the context what is the space where the functions have their values, one often writes $L^2(X) = L^2(X,Y)$.
In your case, we see that it is the space if functions with value in an other space of functions. This is often used in the case of time dependent functions and one has the identification $$ f(t,x) = f(t)(x) $$ And so with these definitions you have everything you need to get the norm of your space.