I have an old computer running Ubuntu 10.04. I installed a LAMP server on it and it works fine; however, the computer is connected to the internet through a large network connected to one modem. I have the ip address of the computer/server for our network, and it works fine, but I need this server to be accessible from the internet, anywhere. I have the ip address of the modem, but using it returns a "server not found" error. Any help would be greatly appreciated.
You need to setup a NAT or port forwarding to redirect a port on the modem to point to your server.
Your router's firewall is probably blocking the traffic. You need to either tell your router to put the server in its DMZ (so it will receive all traffic) or tell it to forward a particular port (such as 80 for http) to the server's internal IP address.
Your "wireless apple time machine" is probably a router. I think you're referring to a time capsule. The good news it has a very nice NAT interface, just download the apple configuration utility. These instructions look pretty good: