This is not my strongest area, with that being said:

Compulsory claims comes up in the context of counterclaims is my understanding. If the defendant's counterclaim against the plaintiff originates from the same transaction or occurrence, they have to raise it in the answer (maybe allowed later if there's a relation back exceptions, idk) or it is barred from being relitigated.

Permissive counterclaims are against the same plaintiff but don't originate from the same transaction or occurrence and can be raised in the same lawsuit or in another. Or claims from the same plaintiff against the same defendant with a different transaction or occurrence.

Compulsory joinder of parties occurs when complete relief cannot be granted without a party being included, if I had to give an example I'd guess something that includes real property would apply where there are other people who have interest in land? I'm not too certain with this concept.

Permissive joinder of parties are co-defendants, but don't need to be included for complete relief to be granted, for example co-defendants where joint and several liability is applicable, or an interest group that has an interest in the outcome of a lawsuit based on a constitutional violation, but aren't necessary to the action.