knowing your IP is next to meaningless. You want mine? I'll give it to you if you want. There's not much you can do with it. If you're desperate, you might trace it to the provider (I have SBC Yahoo DSL) but beyond that there's not much information connected to it.
No one can tell if you're on or offline, if you're logged into AIM or not, nothing. Your IP is just a way for the network to know which computer to send information to, like a mailing address for your computer. Think of it like this, just because you know your friend's phone number doesn't allow you to know when he's leaving his house or making dinner. Same principle.
If you're worried about someone hacking into your computer, even though the chances are minimal, make sure you have a good firewall running. I've got two basic firewalls -- the one provided by my router, and the one provided by my antivirus protection suite. i could install a few more if i wanted, but there's not really a need.
edit: if you want to check your IP address periodically (to see if it is in fact changing) you can use this site:
Current IP Check