your ISP will assign an IP address dynamically - that is, you'll have a different IP address each time you connect. That isn't strictly true, because sometimes some ISPs assign IP addresses to broadband customers on a 'lease' basis, whereby you have the same IP address for a period - a couple of weeks or months. To all intents and purposes you can ignore that, and assume you have a different one every time you connect. The purpose of the address is to identify your computer uniquely for that session, and each time you hit a website the server logs your arrival, and your IP address.
This information can be useful, and your dynamically-assigned adddresses are archived by your ISP, and may be used by the Police to track you down if you are suspected of criminal activities.
If your computer is behind a router then the router will get the assigned IP address, and it in turn will allocate addresses to all machines to which it's networked. These addresses will be invisible to the internet side of the router - all computers will appear to be coming from the same address as far as server logs are concerned.