Generally speaking, border routers are routers that sit at the edge of your network. By that I mean the they are the routers that route between your network running an IGP protocol (RIP, IGRP, EIGRP, OSPF, IS-IS) and an external network thru an EGP protocol (such as BGP), this is common in fairly large environments.
In OSPF, you can also have a router called ABR (Area Border Router). OSPF breaks up Autonomous Systems (AS) into smaller routing groups called Areas. In this case, an ABR is any router that is routing between two or more OSPF Areas. Since OSPF allows you to have many areas within an AS, your friend is correct is saying that you can have a border router within an AS. This case of a border router is very specific to the OSPF routing protocol.
cephas@work 