The router acts as a middleman between your DSL modem and your computer(s). To make a long story short, it will let you run up to 253 computers on your single connection, and unless you specifically set it to reply to probes from computers on the Internet, it will just ignore them and throw away inbound data traffic that none of your computers asked for.
The Netgear I linked to also can filter for words in website addresses. For example, you could block all sites containing the word "micro" and it would block
www.microsoft.com,
www.micron.com, etc. It can also cut off Internet access to all computers on a schedule you set, so your kids cannot get onto the Internet until, say, 6PM when you're home from work.
The router's firewall capabilities deflect intentional hacker probes, and prevent worm-infected computers from infecting your computers with worms
as long as the infected computers are on the Internet. If one computer in your own household gets infected with a worm, however, the threat is "inside" the firewall and now it comes down to whether each individual computer can stave off the attack from its stablemate. To stave off an "inside job," some or all of these would be good:
1) a separate software firewall on each computer, set up to block traffic from the other computers (ZoneAlarm with Trusted Zone security set to High, for example)
2) a fully-patched Windows installation
3) a strong password for all the computer's user accounts, most especially the Administrator-class ones
4) properly-configured antivirus software with up-to-date antivirus definitions, and set up to deal with threats silently, not to come asking you what it should do
5) an Athlon64 CPU with WinXP SP2, which is inherently immune to buffer-overflow attacks
If your kids' computers have Win2000Pro or WinXP Pro, then it would be prudent to set them up as Restricted Users so they can't install software, especially spyware/adware.
There are some more tips and links on security if you click the bottom link in my signature and go to the Resources page. The Microsoft Baseline Security Analyzer is a good one to run since it goes beyond what Windows Update checks on (weak/blank passwords, etc).