They'll be talking via an ordinary phone connected to the standard local telephone network. However, that person's analog voice will be digitised and routed onto your Internet phone company's IP backbone. Your ISP forwards the packets to your PC and its sound system for you to hear. To use an Internet phone service via your PC, you'll need an account with an Internet phone company. The company will usually provide custom software that will take care of account details and charging as well as the call itself.
Some companies bundle special PC phone handsets or microphone headsets with their software.
Starter call credit is usually included in retail product packages to get you going. If all you need is the software, it can often be downloaded from the service provider's Web site. Account creation is done using an on-line form. One disadvantage is that a service like this doesn't provide you with a conventional return phone number. Until recently, there would be no way for someone to phone your PC back.