VoIP stands for 'V'oice 'o'ver 'I'nternet 'P'rotocol. As the term says VoIP tries to let go voice (mainly human) through IP packets and, in definitive through Internet. VoIP can use accelerating hardware to achieve this purpose and can also be used in a PC environment.
Many years ago we discovered that sending a signal to a remote destination could have be done also in a digital fashion: before sending it we have to digitalize it with an ADC (analog to digital converter), transmit it, and at the end transform it again in analog format with DAC (digital to analog converter) to use it.
VoIP works like that, digitalizing voice in data packets, sending them and reconverting them in voice at destination.
Digital format can be better controlled: we can compress it, route it, convert it to a new better format, and so on; also we saw that digital signal is more noise tolerant than the analog one (see GSM vs TACS).
TCP/IP networks are made of IP packets containing a header (to control communication) and a payload to transport data: VoIP use it to go across the network and come to destination.
Voice (source) - - ADC - - - - Internet - - - DAC - - Voice (dest)
When you are using PSTN line, you typically pay for time used to a PSTN line manager company: more time you stay at phone and more you'll pay. In addition you couldn't talk with other that one person at a time.
In opposite with VoIP mechanism you can talk all the time with every person you want (the needed is that other person is also connected to Internet at the same time), as far as you want (money independent) and, in addition, you can talk with many people at the same time.
If you're still not persuaded you can consider that, at the same time, you can exchange data with people are you talking with, sending images, graphs and videos.
Unfortunately we have to report some problem with the integration between VoIP architecture and Internet. As you can easy imagine, voice data communication must be a real time stream (you couldn't speak, wait for many seconds, then hear other side answering): this is in contrast with the Internet heterogeneous architecture that can be made of many routers (machines that route packets), about 20-30 or more and can have a very high round trip time (RTT), so we need to modify something to get it properly working.