The box that plugs into your DSL line (phone line) is the Modem. You need both. You should have a Modem, and a router should plug into that box (not your PC for security reasons) The router will handle multiple connections, provide WiFi, and handle network communications for multiple devices over one connection simultaneously
That is, of course, unless you have a combo 2-in-1 Modem router. Which has some limitations on upgrading those systems as they are produced in limited variations. Best thing to do, would be to find out what type of DSL Modem you need/have, and buy one of those, and then buy a router as a separate piece. Having a standalone router will allow you to upgrade and pick a unit that best servers your needs better.