Suppose an enemy has obtained the IP address of my home computer, and communicated this fact to me in an attempt to threaten/blackmail me.
What exactly are my risks, and how can I minimize them?
My OS is Windows Vista, and my ISP is Verizon.
Your router firewall will likely protect you from anything except a denial of service attack (your vendor may have some defenses against this). Make sure you haven't opened any ports needlessly. Rebooting your router may trigger an assignment of a new IP address.
Contact your ISP and let them know that there's been a threat and to ask them what they can do and what you should do. While law enforcement won't be able to do anything yet, you may be able to file an informational report with them to document what's occurring.
By the way, if this person "knows" that your IP address is in the range 192.168.. or 10...* then don't worry. These are private ranges that are not routable of the internet. Your PC will be assigned an address in this range which can only be reached by the router and other computers, if any, on the same local segment.
Unless you've done something blatently illegal, there should be nothing to worry about.
Most ISPs give you a dynamic IP address which changes occasionally, so your enemy would have to know what your IP was at a particular point in time to be able to trace any activities back to your computer. You can attempt to get a new IP assigned by issuing the following in a command window: