Do you need an external modem and router if your PC has a built in modem?
The word modem is a little ambiguous, because in casual use it can mean any of several different devices. A built-in modem on a PC is almost certainly designed to run over a normal phone line (POTS, or “plain old telephone service”). If you want to connect to a dial-up service provider, the internal modem (well, plus a working phone line that you don’t mind tying up, an account with that service provider, and a phone cable) is all you need. If you want a high-speed (cable or DSL) connection, you are (probably) going to need an external modem of the appropriate type. It would certainly be possible to build a cable or DSL modem into a computer, but I’ve never personally seen one and they’re definitely not typical mass-market equipment. You don’t need a router unless you want to hook up more than one computer, or want to allow a laptop to access the internet in wireless mode. You will need either an Ethernet or USB port on the computer. The internal modem is not used at all in this case.