I am
assuming you are at home. If there is a "box" on the line running from the "jack" on the wall to your computer, it is a router, also called modem/router.
The router itself is basically a device that allows more than one computer in a house to share the same connection to the internet. Employing a router changes the status of your connection from a single user, becoming a "network".
Say, for instance, you and another person in your house have a computer. If you are both connected to the internet, you have your own private network. (I'm trying to keep this simple, and I hope other readers here won't start with the abstracts).
You have a variety of options in this "network". You can both share the connection, with no access to each other's computer, or you can enable "file and printer sharing" to any limit you want.
In a parent - child setup, you might see that the parent has total control of his child's computer - Daddy can be sitting at his own computer, and actually see what his kid is doing, upload files to his kid's machine, download files from his kids machine, restrict/deny access to the internet, or certain sites, and so on, and yet the child can't even "see" or access Daddy's computer.
Or, in the abstract, teh kid could be a real techno-geek and run a sub routine that could make Daddy believe that he has control, while the kid is really surfing all the porno sites, or web - camming with some girlfriends from their machines doing, well, what Mommy and Daddy wouldn't be too happy about.
The funny thing is, Daddy is probably doing the same thing himself!
But I digress. Hell, Daddy could even be in the same -... well, let's not go there.
What was your question? Oh, yea. Routers. At the base level, they simply conrol and direct traffic on the same line (network).
Chances are, if you have hi-speed internet service from (among others), your cable TV company, or phone company, the technician installed a router.
Do you need a router if you are the only person in your house? Depending on your service provider, no. I have FIOS (fiber) service, and I am running straight out of my ONU. (Optical Network Unit - Don't ask, it will just make this reply longer.) Alright, it converts the light beam running thru the fiber into a digital pulse.
I do not know anything about CATV, so I cannot speak for them. With my service, I don't use a modem, I use an ETHERNET card. Well, in a way it's like a phone modem, just like a spear is to a cruise missle. Just a little more Hi-Tech.
ADVANTAGES/DISADVANTAGES USING A ROUTER ON A SINGLE LINE
They can be a real pain in the *** to configure, and innacuarate configurations could kill your connection speeds. If your computer can't "talk" to the router, well, it ain't going out to play. It's just another door you have to go thru.
BUT>>>> A Router is also a "hardware firewall". There is no software firewall that will even compete with the protection it offers... IF YOU HAVE IT CONFIGURED CORRECTLY. A hardware firewall keeps the bad guys from "knocking at your door". They can't even "walk up to your house", because they can't see you.

IF IT IS CONFIGURED CORRECTLY. It's really not all that complicated, really, but this is turning into a seminar.
A SOFTWARE FIREWALL, such as Zone Alarm, Commodo, Windows Firewall

... Symantec,

.... the Mexican Border..
With a software firewall, the bad guys are already at your door. If they can find you. The point is, they are aleady at your door. And if they are persistant enough, they will find a way in. You have to realize that you have like, 65000 doors (ports) to protect. Not including open/hidden sockets, ET phoning home, King George (Bush) and his friends, among others.
I am getting way way
Damn, reading this novel is probably enough to force you to go out and do drugs.
And I probably didn't answer your question.