This all comes under the general heading of VoIP - Voice Over Internet Protocol. There are several different protocols in use by different systems. With the correct hardware/software, you can set up a direct link between any two points with internet access and then have completely free communications. Skype, Vonage, and many other VoIP providers then also provide access to the PSTN (Public Switched Telephone Network) for appropriate fees, allowing you to make calls to any regular number and to receive calls from the PSTN via your internet connection.
The phone number can be based anywhere, even if you are actually using your VoIP service from somewhere else in the world, e.g. you could have a New York number with 212 area code which comes through to you anywhere in the U.K., thus anyone within the NYC local calling area will call you just as if it were a local call.
If you prefer to avoid tying up your main computer and using headsets, you can use an ATA (Analog Telephone Adapter) connected into a router which will then allow you to connect any regular telephone and use it in the more conventional manner. If you leave the router and ATA on all the time, you can still use the VoIP phone when your main computer is shutdown as well.
The main disadvantage to all VoIP at the moment is simply that there is no guaranteed quality of service. You're breaking the audio up into data packets to send over the network, but there is no guaranteed maximum packet transit time. If contention issues result in packets being delayed, audio quality will suffer.