A softphone is basically an application that sits on your computer desktop and emulates a phone through the PC. So if you want to dial a number to ring out, you use the mouse to press the buttons on screen or you use your keyboards number pad. Incoming calls are presented on the screen and you interact with the call (Answer, put on hold, transfer, etc) via the screen, mostly using the mouse or keyboard shortcuts. The audio is routed via the PC using the PC's sound card and a PC headset.
Examples of two of the best ones are
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The benefits of a softphone are that they are generally cheaper, are useful for saving space as extra wires and physical hardware is not needed and can be incorporated into other applications and user network rights.
The problem with softphones is that they present a level of abstraction from the phones functionality that while seemingly fine initially, gets annoying at times. I've never really met an office worker who was happy using one over a real desk phone.
Because the softphone has to be visible on the computer screen for the most part to be useful, it tends to take up valuable screen estate which some people find annoying as well.
I will give you an example. You get a call incoming so it appears on screen. You click to answer and it's someone who wants some information. You then have to hide the softphone to use the browser on your computer to look the information up so that involves a few clicks of the mouse to put the caller on hold, hide the softphone window, use the browser, click to bring the softphone back and unhold the call.
It's really hard to describe why using a softphone over a regular phone is a pain when your blustering around the office in the flow of work but it's very easy to appreciate in action. As I said, I've never met an office worker who preferred a softphone over a real phone. I've even had a few companies install them thinking they can keep their costs down and then have me come back a few months down the line and replace them with real phones.
The people who make the best use of softphones are call center workers who have limited interactivity with the phone and generally have a fixed platform of software on their computers.