Just curious…
Tell us your IP address first…
Knowing someone’s IP address really isn’t as interesting as it used to be. Unless you’re running some open internet-accessing app, there’s not much that can be done with an IP.
You know, the usual, gender, shoe size, penis length or bra size, any type of hereditary diseases.
You can do a traceroute and get a decent idea of a person’s geographic location, often down to the city they are in. This will also tell you who their ISP is, but aside from that there’s little else you can learn about a person from it.
I can tell your mother was a hamster and your father smelled of elderberries.
Here’s an attempt at a serious answer.
If I have your IP address, I know that a computer is at that address. I can tell what ISP that IP address is using. I can probe it for open ports. What that means is:
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If you’re running terminal services or telnet, I can attempt to log in to your computer.
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If your ISP and/or firewall allow netbios traffic, and you have file/print sharing enabled, I can try to access these shares. If you have a stupid password protection, I’ll get it.
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If you’re running any sort of server (ftp, http, sql, etc), I can attempt to break into it.
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I believe that I can send you annoying MS-Messanger pop-ups that can only be blocked by certain archaic rituals on your part.
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If you have accidently installed any trojan horse software, I can use it to take control of your computer.
Firewalls solve most of these problems.
-lv
What QED said. And in addition, your ISP has a record of IP addresses that are assigned to you. If someone knows your IP address as well as the date/time you used it, they can get your name and address from your ISP. Of course they need a court order or warrant to do this.
Q.E.D.? - is there an easy way to do this? Is there a service on the web which will perform this for you? Or do you need a special bit of software to do it?
I use Neotrace Pro, but you can also get Neotrace Express, a free shareware version that does pretty much the same thing. You can download it here.
Thank you! Here’s some good karma coming your Way! 
Karma is fine, but I’d prefer a bottle of Bailey’s. But keep your vegemite to yourself, thank you! 
What QED and LordVor sez.
Search for ‘“geographic location” “ip address”’ and you’ll get lots of hits. Most ISPs and sites include a mailing address in the DNS info (search on “whois”), and your IP will be included in their netblock.
Yeah, you can get ISP, general location, I can make a good guess as to whether you have a static or dynamic connection, helping me get what kind of connection you use (modem, cable, etc).
If you’re on-line at the time, the rest of it follows. I can find out what OS you’re using, and what kind of services you have running (see queso, nmap, and more I’m sure). Those of you w/o those can get some info easily with http://netcraft.com (see this board’s info here), provided the person is running a web server. That’s about all I can think of at the moment. A good firewall, or pretty much any router will keep some of the active stuff away from you, but ppl will still be able to guess your representative. 
I should point out that getting your IP address is easier said than done. It can be extracted from the headers in email you send to me, but unless you have a static IP, that will have changed if you’ve logged off your ISP since then. I can also get it from websites I own or have administrative permissions at when you visit them. Or if you try to access my computer. But getting the IP address of some random poster on, say, this message board would be nearly impossible for me, unless I can get you to email me. And yes, the mods have your IP address. Be afraid.
Standard email headers only show the IP of the mail server that the message was sent through. This will usually belong to the sender’s ISP. You won’t see the individual’s IP unless they’re running their own mail server or are using something like Hotmail which stamps each message with the sender’s IP.
From just your IP adress, without hacking, someone could find out your rough location (the country or maybe even the region you are in) and your ISP.
LordVor and Q.E.D. have got it right on. Be cautious mostly in how you keep passwords.
scr4. Not if the IP’s are dynamic.
Actually, since the majority of ISPs hand out dynamic IPs, then yes, they do keep a log of who had what IP at a given time. So yes, if a court warrant was issued, an ISP would say that yme had this dynamic IP at this date/time.
Which court? Ashcroft’s secret security court which nobody knows about?
Follow-up on the port-scanning thing:
Although a hacker could do any of these things, he most likely wouldn’t unless you gave him a reason to target you specifically, as it’s much easier to just run scripts to port-scan ranges of ip addresses looking for an open server than to make a targetted attack on one IP address.
-lv