well when you are browsing, the IP you present is your ISP assigned IP. So if you are browsing a site which collects 'anonymous' stats such as IP, someone who gains this can then send you NET SEND messages (those annoying windows pop-ups, which you can turn off), run a denial of service attack on you, do anything really.
When you're behind a hardware firewall, each computer behind it has an address which is only valid in your network. All pings to your ISP ip will go to your firewall. You can set up port forwarding, so if someone wants to get to port 80 (http), you can forward them on to a computer behind the firewall (only for that service). You can set this up for any port, and it's useful for all sorts of things from Quake to FTP.
You can also set up a DMZ (de-militarised zone) - which is a computer behind your firewall which has all ports open to the internet. This isn't really recommended, as it leaves you very vulnerable.
have a nice holiday!