Routers and firewalls

Wall Of Flame
What is a "router" or "firewall router" and do I need one?
Short answer: yes, you need one.  Go buy one, follow the instructions that came with it, and set it up.  You shouldn't have to touch it again, except perhaps to occasionally reset it.

Longer answer: a modern router is really (at least) two things together: a device which routs internet traffic to one or more devices on your network, and a "firewall", which prevents incoming access.  Most routers also include a "switch", which connects several devices together, and many are also wireless and allow laptops and other devices to connect to your network with their wireless radios.

Let's take these separately:

Routing takes your network traffic and checks whether it is local or internet traffic.  Local traffic is routed directly to the other machines in your "local area network" (LAN).  Internet traffic is forwarded through the router to the internet.  This allows your network to treat the two types of traffic differently.  Unless you're doing something sophisticated, like hosting a game server, this should all be transparent to you.

A firewall is a very important security measure, and no one should be connecting a computer to the internet without one.  It prevents any incoming connections you haven't asked for from reaching your machine; so, if you browse a web page, that page can come back to your computer, but someone trying to log in remotely doesn't get through. Your PC or Mac should have a firewall built-in, unless you're using a Windows version prior to XP's service pack 2, in which case you should install that right now.  That built-in firewall is a good thing, but much better than that is to have a firewall between all of your traffic and the internet.  Both are important; nothing is foolproof, and even having a firewall in your router won't prevent one of your computers from getting infected all the time.  Those built-in firewalls will protect you to some extent when that happens.

The switch aspect isn't really anything you need to worry about, but it's what lets local network traffic get from machine to machine if you need to.  It's also transparent if it's working correctly.

The wireless access is an optional feature that makes your wireless router also a wireless access point.  If it's included, it will have its own setup instructions.  Wireless devices have gotten super cheap these days, and home users who aren't running something like their own home media server don't need an expensive one.  As long as you're buying new off the shelf, it will be fast enough that you won't notice the difference between wired and wireless access; modern wireless devices are able to move data much more quickly than your cable modem, so they'll never be the bottleneck.

If you're looking for a home router, Belkin and Linksys (which is now part of Cisco) make good ones.  For business use, because your data is more valuable you want something with capabilities that I haven't discussed here.  SonicWALL makes an excellent business-class product that, when properly configured, meets HIPAA and other security requirements.