Your IP Address is assigned by either your ISP or Network administrator, or yourself (in case of home network.) The ip address can be run through whois, nslookup etc.. which will return general information. visit www.arin.net and put your ip address in their whois box. It will show who owns the ip, and possibly their information.

One thing to keep in mind is that the ip address you are assigned when you connect to the internet does point at your computer or in my case a wireless router. So, it's possible to launch port scans to see what ports are available on your computer, and possibly exploit them. A firewall, such as WindowsXP built-in or ZoneAlarm, or Norton Personal Firewall can certainly diminish the problem.

If you own the ip address, and have a domain name attached to it, then it may be possible to get the registrar contact information from it. This is not typical though, and if you bought the ip address you would know.

ISPs typically do not associate any kind of user information with the ip address, so if someone is claiming to have looked up your information based solely on your IP, then they are b-sing you. However, if they found an open port on your machine they may have been able to gain access and retrieve information that way.

I'll second sfbell's assessment. Although it's a relatively common boast in chat rooms, that kind of "I got your IP, now know all about you" drivel is BS, for just the reasons sfbell explained.