If you have an Internet Protocol address, you can find some information about what internet provider it's associated with and roughly where the user is located. In some cases, this information may be misleading if a connection is being routed through multiple computers or a virtual private network. You generally can't get information precise enough to trace an IP to a home address without working with the internet service provider in question.
How IP Addresses Work
IP addresses are numerical codes that are used to route traffic across the internet, somewhat similar to phone numbers. Computers use what's called the Domain Name System to translate human-readable addresses, like www.example.com, to machine-readable IP addresses like 192.168.0.1.
Just as it's often possible to tell roughly where someone is located based on a landline phone number, it's often possible to tell someone's rough location using an IP address, since they're divided into groups to make it easy for computers to know where to send particular traffic.
IP Geolocation Services
You can find various IP location services online that will tell you roughly what city an IP address is located in and what internet service provider it's associated with. In some cases, the provider may be an organization with a well-known site, such as a large employer or a university.
Generally, you can't get an address from an IP, since the publicly available data simply isn't precise enough. There's also always a possibility that the information you get will be wrong, if databases of IP addresses become out of date or a connection is routed through multiple computers in multiple places, such as when someone uses a VPN.
Getting Address Information From ISPs
Internet service providers generally do store what IP addresses were assigned to which subscribers at what time, but they usually won't hand out this information without a court order.
In some cases, if you're suing someone for defaming you, infringing on your copyright or otherwise violating your rights, you may be able to get this information through legal channels. Police can also get IP address information if a crime is suspected.
If you think a crime was committed and an IP address you know is relevant, contact the police. Consider contacting a lawyer if you think you have a civil case against someone and know his or her IP address.
Note that ISPs will know what subscriber was assigned to an IP address, but that won't tell you who is actually personally responsible for any online activity. Wi-Fi connections can be hacked or used by neighbors without permission, and houseguests or workers might have access to a Wi-Fi password without the constant supervision of whoever pays the internet bill.