Internet telephony, or more common, Voice over IP (VoIP) or digital telephony allows parties to exchange voice data flows over the network. The big difference is that the data flows over a general purpose network, the Internet, contrary to conventional telephony, that uses a dedicated network of voice transmission lines. The two networks can be connected, however, under special circumstances, but for now this is certainly not a standard. In other words: it is very likely that you will not be able to call people who are using a conventional telephone. If it is possible at all, it is likely that you will need to pay for a subscription.
While there are currently various applications available for free download, both free and proprietary, there are some major drawbacks to telephony over the Internet. Most noticably, the system is unreliable, it can be slow or there can be a lot of noise on the connection, and it can thus certainly not be used to replace conventional telephony - think about emergency calls. While some providers take their precautions, there is no guarantee that you can reach the party that you want to call.
Most applications currently do not use encryption, so be aware that it is potentially easy for someone to eavesdrop on your conversations. If security is a concern for you, read the documentation that comes with your VoIP client. Additionally, if you are using a firewall, it should be configured to allow incoming connections from anywhere, so using VoIP also includes taking risks on the level of site security.
First of all, you need a provider offering the service. This service might integrate traditional telephony and it might or might not be free. Among others are SIPphone, Vonage, Lingo, AOL TotalTalk and many locally accessible providers offering the so-called "full phone service". Internet phone service only is offered by Skype, SIP Broker, Google and many others.
If you want to set up a server of your own, you might want to look into Asterisk.
On the client side, the applications that you can use depend on your network configuration. If you have a direct Internet connection, there won't be any problems, provided that you know on what server you can connect, and usually that you also have a username and password to authenticate to the service.
If you are behind a firewall that does Network Address Translation (NAT), however, some services might not work, as they will only see the IP address of the firewall and not the address of your computer, which might well be unroutable over the Internet, for instance when you are in a company network and your IP address starts with 10., 192.168. or another non-routable subnet prefix. This depends on the protocol that is used by the application.
Also, available bandwidth might be a blocking factor: some applications are optimized for low bandwidth consumption, while others might require high bandwidth connections. This depends on the codec that is used by the application.
Among the most common applications are the Skype client, which has an interface that reminds of instant messaging, and X-Lite, the free version of the XTen softphone, which looks like a mobile telephone. However, while these programs are available for free download and very popular, they are not free as in free speech: they use proprietary protocols and/or are only available in binary packages, not in source format.
While your computer, especially if it is a laptop PC, might have a built-in microphone, the result will be far better if you connect a headset. If you have the choice, opt for a USB headset, as it functions independently from existing audio hardware. Use alsamixer to configure input and output sound levels to your taste.
VoIP applications are definitely a booming market. Volunteers try to document the current status at http://www.voip-info.org/.