Codec stands for Coder/Decoder. Basically it is a piece of software or a driver that adds a support for certain video/audio format for your operating system. With codec, your system recognizes the format the codec is built for and allows you to play the audio/video file (=decode) or in some cases, to change another audio/video file into that format (=(en)code).
For example, when you install Windows to your home computer, Windows installs automatically bunch of most commonly used codec into the system, so you don't have to download them separately from their vendors. Despite that, there are some codecs that are widely used, but not installed automatically by Windows -- most notably DivX ;-), MPEG-2 and in some cases (old Windows versions, like Win95) MPEG-1 codecs.
To download the most commonly needed codecs for your computer, check out our list of audio and video codecs for Windows.