The "C" or Common wire is used to get 24VAC constant power to the thermostat.
There are an increasing number of thermostats that require this as they get more advanced and require more power; thermostats with WiFi and/or touchscreens almost certainly require a "C" wire. Older programmable thermostats typically used batteries to provide power, but as most had a simple LCD screen and relatively simple timer electronics, the batteries could last for years. Many of the new thermostats will run on batteries, but this is intended as a backup; a WiFi radio draws enough power that running on batteries alone it will only last for a few days.
It is still not common practice for HVAC installers to run a C wire to the thermostat, and if your home was built pre-2010, chances are high you don't have one.
If you need a C wire, before you ask...
- More than you ever wanted to know about the C wire outlines what all the wires are, and how to get a C wire to the thermostat.
- Browse through questions in this tag, there are many specific situations covered, especially dealing with older furnaces and wiring.