A modem gets you connected to the internet. It translates signals sent over the phone line into data that your computer can use.
A router can take that connection that a modem creates, and share it with (or route it to) more than one computer.

You need a modem to get on the internet. You need a router to use that connection on more than one computer at the same time.

I'll venture the additional.

Modems come in at least three "flavors"

Voice-band or dial-up modem for use over regular telephone lines, this is the "modem" we have been using for the past 30 years, peaking in the 56KBits modems meeting the ITU standards V.92

Digital Subscriber Line (DSL) modems that "push" the regular telephone lines to give rates as hight at 3 Mbits and allows the line to be used for voice telephone at the same time.

Cable modem that connects to the tv cable. This may be much hight yet, maybe 10 Mbits, I don't know the cable modem rates, I use DSL and Dial-up.

A Google search using the words computer definitions provided numerous items. Included was a source often referred to here at the Forum.

DA

Here is a link that might be useful: Webopedia

Thanks for all the help.

If I only have one computer, then I would not need a router?

Would there be any other reason to have a router? Does it protect the computer in any way?

Thank you Damccoy for the link. Sometimes those definitions use other words I don't understand, and that's why I asked here for the simplest version. I will save that link, tho.

Yes a router normally has a built in hardware firewall which definitely helps keep your pc more secure.

I have verizon fios (fiber optic internet) and I have a dlink router that my ethernet cable plugs into straight out of the wall, then from the router I plug in an ethernet cable to my various pcs, it serves as modem and router.

a router should be set up with the firewall active, and it should be set up with a good password not left with the standard one that came with it.
A Nat router is a hardware firewall where as programs like zonealarm are software firewalls, I always recommend using both especially with any type of broadband connection.

If I had only one computer I would not add a Router. I ran that way when I used dial-up from two computers, and of course that meant only one computer on-line at at time. When I purchased DSL service I also purchased a Router, as there's no other way I know of to connect two or more computers to one DSL modem, nor is there a way to put more than one DSL modem on a single phone line. Sorry if this got over complicated, just couldn't stop typing. Again, you do not need a Router if you are connecting only one computer to the phone line (dial or dsl) or cable.

Most high-speed isp modems will have a router already built in. One fairly easy way to determine is maybe go here and see if it shows that you have an internal LAN ip number.