VoIP stands for Voice over Internet Protocol (also commonly referred to as IP Telephony). It is an alternative way of making calls, and does not use traditional landlines, but rather the infrastructure of the internet. VoIP calls not always include the use of a phone either, as many VoIP calls are made using computers.
VoIP has become particularly popular for long-distance calls, mainly due to the greatly reduced cost. However, most people who use VoIP use it to connect to other users with VoIP, in which case using the service is completely free. The most common way people do this is by using software such as Skype installed on a home PC with a microphone.
There are two main ways to use VoIP: it can be used to connect to another VoIP service such as a VoIP-enabled phone or a computer. The other way is to use VoIP to connect to a standard landline, in which case a cost will be incurred (although often cheaper than connecting two landlines, particularly for long-distance calls).
The great thing about VoIP is that it uses already existing infrastructure (of the internet) to make voice calls. VoIP transmits sounds over an internet connection, using the Internet Protocol (it's name, Voice over Internet Protocol). For this reason, VoIP calls between two people with VoIP are often free.
VoIP is not free if you want to connect to someone with a standard phone-although the cost is still a lot cheaper. If you wish to have a VoIP enabled phone at home or at your work place, you'll require a paid service with a company such as Vonage or Lingo. These paid services make a lot of sense because the calls are still a lot cheaper as opposed to a standard landline.
At the moment, VoIP is still a reliably new technology, although its rapid growth can be compared to that of the internet in the late 90's. In the next couple decades, we can expect VoIP to replace standard telephone systems all together.