As mentioned in other posts, assuming your "router" is a NAT router, you should be relatively safe from any attacks from probing from the internet. However, as also mentioned, the machines on your LAN are still vulnerable to attacking each other if one of them gets infected by a virus.
I don't feel that there is any need for a 3rd party firewall. Most modern operating systems (Windows XP SP2, Windows Vista, Windows 7, most Linux distros, Mac OS X) ship with firewall which is on by default. That alone should be enough to protect you from any attacks from infected machines on your LAN. I wouldn't worry about installing a 3rd party firewall unless you requires special features such as filtering outbound traffic.