What exactly is a language in Predicate Logic?

Question

I know this is a bit of a dumb question, but it is a source of minor confusion that I have yet to shake off. When we speak of a language $\mathcal{L}$, what exactly is $\mathcal{L}$?. I know we speak of the similarity type of $\mathcal{L}$, which determines our alphabet and thus the constants, predicates, and functions we will be working with on top of our variables, connectives, etc. But is $\mathcal{L}$'s alphabet a subset of $\mathcal{L}$ or a separate thing? Are $\text{TERM}_\mathcal{L}$, $\text{FORM}_\mathcal{L}$, $\text{SENT}_\mathcal{L}$ subsets of $\mathcal{L}$? Is $\mathcal{L}$ the set containing the entirety of the alphabet's symbols and all possible (or perhaps only those well-formed) strings of those symbols, in particular those we deem terms, formulas, sentences, and so on? Or is $\mathcal{L}$ just the set of well-formed strings (formulas) with $\mathcal{L}$'s alphabet? This seems the most likely option, as I have seen plenty of references (with regards to theory-extensions, for example) to a set of sentences' intersection with $\mathcal{L}$. This confusion extends too, of course, to the nature of language in Propositional Logic. Is $\mathcal{L} = \text{PROP}$ (within the context of Propositional Logic)? I thank any help and responses in advance, I know this is quite the silly confusion to hold, but I have yet to fully know the answer.