Just because the computer picks up a wireless signal from a router, it doesn't mean that you have ''connected'' to a the router.
You need to:
1) make a connection to the router (either wireless or wired)
2) have the router connected to a modem with paid Internet service
You or your mother needs to subscribe to some sort of broadband Internet service (Cable, DSL, cellular, satellite, FiOS, etc). The ISP will provide you with a broadband modem for their type of service, or you can purchase your own modem compatible with the Internet service that you purchase. At that point, you can connect any computer directly to the modem to access the Internet. I do, however, strongly recommend using a router for added firewall security.
Internet ----- broadband modem ----- one computer
Internet ----- broadband modem ----- router ----- many computers
Some ISP's will provide one device that is a combination broadband modem-router.
Internet ----- broadband modem-router ----- computers
The router is used to connect more than one computer to form a local network that, in turn, is connected to the Internet. It is an easy way of sharing Internet service among more than one computer.