How does VoIP Work?

VoIP as you know, is VoIP Over Internet Protocol, saying that- we can safely say that- its not much different from normal data which flows in internet. The voice (which is analog) is converted into digital format and put into data packets and sent across internet. Unlike analog telephony VoIP telephony is packet based ( that's obvious).But there is a difference in protocols used for VoIP.
VoIP signalling is mainly done through SIP ( session initiation protocol) and normal data transfer is done through RTP . SIP is not the only signalling protocol used though, it is mainly a session initiation protocol and does the dynamic maintenance while the call is ongoing.Since,most of the computers don't have a static IP address when connected to internet and since many of them use Firewalls- it rather becomes a norm to employ a mediation server. Hence, there is something called as a SIP server or a location server which has the existing mapping of the user name/phone number mapping with the current IP address mapping with the user name .For this, the users needs to be registered. for this purpose the users are registered as say #-Link-Snipped-#.
When ever user initiates a call to another VoIP user, the call is first directed to a location server which locates where the existing end customer is located along with the Ip address and starts forwarding the data packets. It constantly updates any changes in the IP address as well with associated user name.
As you know, this is more of a over view. Internet has quite a few resources where you can refer to.

Just to add to Durga's explanation, the main protocols used in VoIP these days are SIP and H.323 (see comparison here: #-Link-Snipped-#). In addition to these protocols are codecs used, each with their own uses. The G.711 codec is used for high quality calls but require more bandwidth, while the G.729 codec requires much less bandwidth though quality is degraded somewhat.

You can also use a normal telephone for VoIP calls, provided a ATA device is used. Once it is configured remotely (via it's IP address), it can be used standalone.