Videos commonly are compressed, meaning its been altered to take up less space on your computer. For example, a regular Blu-Ray video usually takes up around 30 or 50GB of space—which is a lot. So, we compress movies to make them more manageable, usually with loss in video quality.

A codec compresses and decompresses data. It interprets the video file and determines how to play it on your screen. Your computer comes with many codecs pre-installed, though you can install codec packs for wider support, or a program like VLC or MediaPlayerLite.
Example codecs are:

FFmpeg (which includes formats like MPEG-2, the format in which DVDs are stored, and MPEG-4, which is the video format Apple uses in the iTunes store)

DivX, which works with a certain type of MPEG-4 file, and was often used to rip DVDs in the pre-HD era

XviD, an open source version of DivX, popular among movie pirates

x264, which compresses H.264 videos (Also known as MPEG-4 AVC), and is very popular for high definition videos

There are a lot of different codecs out there, and it can get really confusing with all the different versions of MPEG standards.