When you send any information using the internet one of the pieces of information sent is your IP address. This really is like your return address in the sense that when you request a web page then the page has to be sent back to your computer for you to view it.
Your IP address is nowhere as specific as your mailing address.
There are tools on the internet that say they can trace the area to which your IP address belongs. http://www.hostip.info/index.html or http://cqcounter.com/whois/ etc. However, the accuracy is always suspect. I've just tried my IP address in a couple, one got it right (to the town I'm in), another was off by 200 miles and a third was out by over 700 miles - it couldn't even get the right state let alone town.
Getting it down to the town you're in is the best any of these tools can do. The only people who can accurately match an IP address to an actual street address is your ISP. The only people who will have access to this information outside your ISP are law enforcement agencies and then only with a subpeana.
It is possible that someone with a grudge on a fourm or whatever that posts your IP address can mess up your internet connection by continuously "pinging" your IP address. Basically, they flood your connection and so slow it down. This is known as a denial of service attack (DoS) amd is extremely rare against an individual.
Another thing that can happen is that someone may decide they are going to try and hack your IP address. This is even rarer than a DoS attack and is usually defeated by using a strong password on your computer - http://www.microsoft.com/protect/yourself/password/create.mspx - , using router, using a firewall - http://www.zonealarm.com/security/en-us/home.htm?lid=en-us - and so on. This really is a lot rarer than some people on Yahoo Answers seem to think it is.
Now, one thing that gets said a lot is never, never, never give your IP address to anyone. Well, here's mine 71.114.160.207. Here's Yahoo's - 206.190.60.37 and here's Microsoft's - 207.46.232.182
Generally that's a pretty good rule, it lessens even the small risk of anything bad happening even more, but a lot of forums and other sites log and show them with nothing happening.
I'm no computer security expert by any means but hopefully I know enough to keep my computers safe. Especially so as I run a web server and my IP address is on thousands of computers - it needs to be on the DNS servers otherwise no-one would be able to get to my web sites.
In fact, because of the web server I keep logs of everyone who tries to break out of the web file system - you can see some of these logs about half-way down http://brisray.com/comp/cwserver2.htm