An IP address is sort of like a phone number or street address. It's your address on the 'net.
Since you access the internet via a modem, you get a different IP address every time you connect to the internet.
All internet communications are done by IP addresses. Even when you type in a website name, your computer checks a list to see what that website's IP address is before it tries to go to it.
I don't know about the firewall thing, except that most file-sharing programs, like Kazza, Gnutella, and WinMX will generate false alerts on firewalls unless you configure your firewall correctly.
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