You are a bit confused with the various components required for Internet connections.
If you have a dial-up
connection with an Internet Service Provider, all you do is connect your computer's modem port to the phone line and dial the phone number of your Internet Service provider.
If you are connected to the Internet Servise Provider by ADSL and have a combined ADSL router and modem then you connect the phone line to this router/modem. If the ADSL modem is separate component then the phone line is connected to the ADSL modem and this modem is connected to the router with a LAN cable and the computer is connected to the router with another LAN cable. If the route is
wireless and the computer has a wireless card then you don't need a LAN cable between the router and the computer.
If you are connected to the Internet by ADSL, to secure your Internet connection and configure your computer so it can connect to the Internet, please follow my instruction - click on this link :-