DR TCB - as I understand....the modem is the little box you get from the Internet Company....a router is an in between you can use if you have more that ONE computer or if you want to have a wireless signal....although I think you can also get a wireless modem.
I have my modem from Comcast....which when I only had one computer, I plugged it directly into my computer. Then I got another computer, so I bought a router....I plugged my modem into the router and then ran two cables one to each computer.
I then bought a WIRELESS router so that I would have a signal for my tablet and/or telephone in the house....so I have two computers plugged into the router....which also gives me a wireless signal that I can choose for my tablet,etc.