Once again there's not necessarily hard lines between them and names are sometimes invented to justify something.
However, a basic trainer in most cases will simply be a small, single engnie (piston) airplane, like a Grob 120A maybe, that teaches the basic aspects of flying in a slow mover. Stuff like the consequences of applying bank and / or pitch, trimming, etc. And also showing the general reaction of an airplane to certain controll inputs. And then the basics of comms, a traffic pattern, approach to stall indications and so on.
After the first few flights, the student may progress to a more powerfull turboprop trainer, like a PC-7, T-6A, etc. You can learn a lot of stuff in such a plane that's still affordable to operate, but allready has quite some performance. You can do acrobatics and some more advaned aircraft handling.
Finally, the advanced trainers, like a T-38C, M346 or T-50 will indeed display the performance and handling characteristics of fighters. They have jet engines, swept wings and some proper power. A combo that make for a different kind of flying.