Here is the stub of an answer. Many conjunctions can be used in two or more different ways. And I've only given an example for each category, not an exhaustive list. But this should be enough for you to be able to categorise other conjunctions.
Copulative/Additive: connects two clauses without indicating any specific kind of relation between them: et.
Adversative: indicates opposition: sed.
Explicative: explicates or elaborates on the previous clause: quod/ut "that".
Final: indicates a purpose or end: ut, ne
Causal: indicates a cause: nam "for", quia, quoniam, quod "because".
Comparative: indicates what the preceding is compared with: quam "than".
Conditional: indicates a condition: si.
Consecutive: indicates a result that was not specifically intended: ut.
Concessive: indicates a concession: quamquam, quamvis, etsi, ut/cum "although".
Temporal: indicates a time at/after/before which the main clause happens: ut/cum/ubi "when", postquam, priusquam, antequam, dum, donec.
Modal: probably indicates circumstances or methods, but I can't think of what those would be. Perhaps (sic)ut "(such) as", but one might as well call that comparative. Or prout "to the extent that".
Local: indicates a location: ubi "where*.
Disjunctive: separates two possibilities: sive, aut.
Interrogative: introduces the content of a question: utrum, an. Or perhaps others would have this fall under "explicative".
There is no necessary or complete list of different kinds of conjunctions: any categorisation is a choice. But most of the terms above are indeed fairly commonly used for conjunctions.