A router is designed to link up multiple "sites" across a WAN (Wide Area Network). That would be you as "Site A" and your Internet Service Provider (ISP) as "Site B". Now if you want to hook up more devices (or just make jacks live) at one of these sites, you would add a switch (old days it was a hub) to your router. This essentially adds more ports for you to plug in to (won't go into detail about what a switch does exactly, because as far as you need it, you know enough now).
Now, just follow the other guys' advice and you will be all set.
That is basically what I have. My internet comes in my house via cable. Goes to my cable modem, then to my Vonage router. Now, I do have another router in my mix, but it is for wireless, so we'll ignore that for now. Then I have multiple switches feeding the approximately 38 Cat5 network lines in my house (talk about a lot of wires). It was cheaper to pick up five 8-port switches and daisy chain them (they had dedicated "uplink" ports so that I could still use all 8 switch ports for my network runs) than to buy a single switch with enough ports to support 38 lines.
Now, if I could just get all of my cable TV runs into a powered distribution box of some sort, I'd be set (I have 1 cable run for each network run and for each phone line run). To do that would cost a small fortune. The phone lines aren't important because we use cordless phones.
Good luck!
