Is ‘that’ a relative pronoun?

Yes, that is normally classified as a relative pronoun when it’s used in a phrase like ‘the house that I live in’ or ‘the train that departs at 4 p.m.’

Some modern grammars disagree, classing it as a subordinating conjunction instead in such cases, but traditional grammars usually call it a relative pronoun.

In other contexts, that can also function as a conjunction, adverb, or determiner.