Well, I neeed to know how to get the dinamic ip of a computer, because I'm making my Own messenger,wel not that good, but i found a difficulty when I tried to connect to my friend's computer in the internet.

he MUST be connected to a router AND MUST have a port routed.
There is no other way you can contact him WITHOUT server.
I have already tested it.

Only works in LAN without routes, servers, etc.

If he goes to http://checkip.dyndns.org/ that'll tell him the IP address you need to connect to. If he's behind a router and/or firewall, they'll need to be setup to forward incoming connections on the port you're using to his computer.

I've read information about NAT, it is a way to connect out of an intranet a stablish a direct connection, but I didn't find any good example of how use it

You don't use it, its all handled automatically by the router when you're talking out through it.