As its name implies, it is a piece of software, or part of a VoIP software, that acts as an interface allowing you to dial numbers and carry out other phone functions using a screen (that of your computer or PDA) and your mouse, keyboard or keypad.
Commonly, the interface resembles a phone with buttons representing the keys, which you press using the mouse to dial. An example is the Skype softphone.
For speaking and listening, a headset and microphone suffice. You can also use an IP phone or handset, if you have one.
Here is a list of the most common popular softphone applications and services.

