Updated July 2013
As of this update, Google Voice is the best way to make free Internet phone calls.
The Google Voice service is more of a "phone number management" service than a typical Internet phone service so it does a lot more than let you make free phone calls. Still, if all you want to do is to make free phone calls, Google Voice is your best bet.
Google Voice was formally known as GrandCentral.
Types of Free Calls With Google Voice:
Google Voice can be used to make free PC to phone calls and free PC to PC calls.
The easiest and most common way to use Google Voice, however, is to allow the service to connect your phone to the destination number for free. I suppose you could call this a free "phone to phone" call.
Sign Up for Google Voice:
You can sign up for Google Voice for free here using your Google Account, the same information you use to access other Google services.
Click on the Get a Google Number button to get your free phone number so you can start making free phone calls!
Google Voice is only open to those in the US.
How To Use Google Voice:
To use Google Voice to make free phone calls, log in to Google Voice and click the Call button. Enter a US or Canadian number in Number to call, choose a Phone to call with (this is your home or mobile number you told Google about when you signed up), and then click Connect. Google will dial the number you entered and connect the call to you for free.
To use Google Voice to make free PC to phone calls, install the free Google Voice and Video Chat Plugin, also called the Hangout Plugin. Once installed, you can make free calls from the Chat or Hangouts area in Gmail to real phones using just your PC microphone and speakers.
PC to PC calls, usually called voice chat or video chat, can be made using Google+ Hangouts.
You can also use Google Voice as a web app from mobile browsers, as shown in the screenshot above. Official Google apps are available for Android, BlackBerry, and iPhone users.
Google Voice Free Call Limits:
Free calls made with Google Voice are limited to three hours in length, at which point you'll be disconnected. However, you're free to call the same number over again as many times as you like so this is more of an occasional annoyance than a restriction.
Also important to know is that calls with Google Voice are free only if you call numbers based in the United States or Canada. Calls to numbers outside the US and Canada will incur a small per-minute fee.
PC to PC calls using Google+ Hangouts are not restricted.
Google Voice Review:
I love Google Voice. I've used it personally since it was GrandCentral, before Google bought the company. The free phone calls are clear and it's a very easy service to use.