Significance of morphisms out of final objects

I read this theorem (Lawvere's theorem) that talks about morphisms out of final objects and I'm wondering if I understand what that means.

I think for sets and topological spaces this is just a set or space with a distinguished element or point.

For groups and rings and modules I don't think it's anything significant.

Do I have this right? Are there other categories where it means something?

Thanks

Author: BananaSmoothy420