Generally speaking there are two different types of compression: lossless and lossy. Lossless compression means that no data is lost during compression while lossy compression means that some loss of data will occur. Thus, when you decompress a file that has been compressed using lossless compression, you end up with the exact same file as you started out with. Text documents and programmes are compressed using lossless compression, because they will otherwise become unreadable or unexecutable. Lossy compression is used for compressing media files. One of the best known compression methods for video files is DivX. When a video file is compressed using DivX, some of the data that the video file contains is lost and the quality of the video file is decreased.
Mp3 is another compression format that uses lossy compression, but this type is used for audio files instead. When compressing audio files using mp3, the bitrate is decreased and the quality of the sound that the file contains is therefore decreased. When a video file is compressed, it is often converted into an AVI file. AVI means Audio Video Interlaced. When a video file is interlaced it means that the file contains a bit of video, then a bit of audio, then a bit of video and so forth. This dramatically decreases the strain on the hard drive because it has to play the video and audio tracks simultaneously, but doesn't have to search for very long for the next bit of the respective tracks. The video and audio tracks can then be compressed using DivX and mp3 (or other codecs), making the file smaller.
You may ask why it is preferable to compress a video file when the quality is decreased. Well, whether you want to store your home videos or backups of your DVDs on your hard drive, you will quickly find yourself in need of extra hard drives if you do not use compression. You also have to take into account that your TV doesn't have the same resolution as your computer monitor, so it is possible to use lossy compression and still have an acceptable picture.