A enables your computer to connect to other computers or to the Internet?

To connect a computer to other computers (i.e. to a network) you need a network adapter, also called a Network Interface Card. This adapter should be suitable for a network which you want to connect to. Wireless adapters are for wireless networks, wired adapters are for wired networks.
A network connection.
if its file sharing a local network over your lan, or if its a game, social networking site on the internet its the internet.
A network card connnects to a router (wireless or LAN) the the router connects to other computers or a modem (intranet-telephone interface)