In a nutshell (oh, spare me the nutpicking!):
NAT - If you're behind a router, it helps passes data back and forth. Without it, data coming your way would end up slamming into the router. Dead.
UPnP - Magic. Automatically configures forwarding of ports (only if your software supports it).
TCP & UDP - Two different methods of sending data around. You should check which your program uses, and if adding the forwarding manually, make sure you know which (or just add both on the same port for good measure).

As for the tracker problems, you shouldn't be getting any issues with it if you can browse websites - the BT tracker protocol runs on HTTP, for the most part. And since those trackers usually maintain good uptimes, you know whose fault it is..