You may want to check out the networking FAQ'a at www.broadbandreports.com - good info in the glossary regarding differences between switches and routers.
A typical home configuration goes from the broadband modem to a router. Hanging off the router can be computers, switches, or other routers. Routers and switches are independently powered. They don't depend upon any one computer being up.
Think of a router as having intelligence to direct traffic, resovle Network addresses (NAT), and monitor who's up/down. A switch is more of a passive device that has limited intelligence - you can think of it as a box that allows you to add ports to your router.
If you get a router that has more ports than you can immediately use, put a piece of tape over the unused ports - they collect dust pretty quick and when you finally do use them, you may find that either the port doesn't work, or traffic through that port is slower due to re-transmissions. - Same applies to switches.
Most HomeNetWork routers have harware-level firewalls. Sometimes these are called "Gateways" - I use a gateway router that has ports for ethernet, HPNA (uses internal phone wireing), Wireless, and USB-eternet.
Off of the ethernet port, I have a another router and also a switch. Computers hang off of each.
Off the WAP (Wireless Access Point) port, I have a WAP router. that is a separate sub network.
The USB-Ethernet port is used occasionally by people with laprtops.
I used to use the HPNA port to access a computer in a room where running cable would have need hard.
So, depending upon your needs, you can configure a network pretty much anyway you want.