According to Wikipedia, a codec is a program capable of performing encoding and decoding a data stream or signal. In English, that means that a codec is a thing you install on your computer to allow you to watch certain types of videos. This means there are lots of different codecs for all the different formats of movies out there. What makes a codec sexy is its ability to produce an ever increasing quality video with a smaller file size. (Otherwise known as video compression)
Fairly recently, a new standard in video compression called H.264 has been developed. It uses the .mp4 extension. It is now a very popular choice with the iPhone, iPod and other mobile devices. I compressed a 1 minute of 720p video (originally 68MB) down to 8.4MB (settings of 640×480 at 30fps,and 40kHz aac audio) and the quality was fantastic. H.264 is my new best friend. I will post some video here soon. I recommend using Quicktime or Media Player Classic (part of the K-Lite Codec Package) to watch H.264 video as opposed to other video players like VLC. (The way VLC Player works is different without the normal codecs, and in the case of H.264 files I have viewed, the images are [unfortunately] significantly darker and less visible.