'He apologised to me for that.'
I know that 'he' is the subject and 'apologised' the verb but I don't know about the rest.
"He" is the subject (pronoun, third person, masculine). And "apologised" is the verb, it is the past (or preterite) form of the verb "apologise". This verb is intransitive, it doesn't have a direct object.
Then there are two prepositional phrases, these are adjuncts (optional elements): First "to me" tells us the recipient of the apology, you might describe the function of this prepositional phrase as dative. Then "for that" tell us the cause or purpose of the apology. Each of these prepositional phrases has a preposition and its object.
I'll include an alternative analysis. "To me" and "for that" are both prepositional phrases, but can by analysed as arguments or complements of the verb. Some verbs allow for optional arguments (for example some verbs can function both transitively and intransitively) But the optional arguments are complements, not adjuncts. In this analysis the verb recipent of the apology is an argument of the verb phrase.
The "relative clause test" is sometimes used to distingish arguments from adjucnts. Many adjuncts can be removed into a relative clause:
He ate a burger with a fork = He ate a burger, which happened with a fork.
So can we do the same thing? Can we rephrase "He apologised to me for that" as
He apologised for that, which happened to me.
If you judge this to be a valid rephrasing of the original the you should call "to me" an adjunct. If you consider this to be unacceptable then you can say "to me" must have been an argument of the verb. You can repeat this test with "for that".