Quote from DElyMyth
If you can reconfigure your internet router, you can install the server on one computer, then reconfigure the router to allow traffic from outside to the minecraft server.
I recomment using whitelist for that (you never know, someone can still get into your server).
This will be your best bet if have any technical know-how.
If you can't, then you can pay for a server with one of the many server providers (mcprohosting, cubedhost, etc). In a few moths Mojang will also release an official server hosting program called minecraft realms.
If you don't have money, and you aren't good with computers you can bring your computer over to your buddies house. This will work well if you have a laptop of course. Then, you can play on a LAN (local area network) server, which doesn't need any router reconfiguration to do.