An IP address is kinda like your real address- it identifies where you are. Sort of. When I look up an IP address for a residential connection, it’s generally going to terminate at the nearest central office. That means I would have a general idea of where you live, which is a great starting point when you’re looking for someone. Exceptions to this rule inlude: cell phone 3g or 4g connections (I live in IL, and I’ve had my IP on my phone terminate in Oklahoma, Ohio, Nevada, and Indiana, depending on the day), along with server addresses (if you use a SSH tunnel), or proxies.
Now, this doesn’t magically give you ALL the information you’ll need. You’ll either need a court order, or some diligence to make sure you get the rest of the info you’re looking for, but still.
Is it illegal to look up an IP address? Not usually. If it connects to you (i.e. your blog, your website, your forum, your server), you’re A-OK to harvest that information.
What do you do when someone flips schnitzel and gets upset that you have their IP?: Generally, I wouldn’t give a shit. Harvesting data you control is totally legal.
What can you do with an IP?: Lots of things. A few of which are which are illegal. Go look that up for yourselves.