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How does data compression work?

If a file is containing info from A to Z. How does compressino actually help reducing the filesize while retaining the same A to Z content?

My best guess is 'compression' works by identifying and removing redundant and repetitive patterns in the file instead of letting them fully take up space...

I just pulled that one out of my ass. How does it really work?
 
We used to use a operating system which (although it wasn't true file compression) used a storage scheme called RAD50 that stored three characters per byte. You gave up some special characters out of the ASCII character set, but it really saved on the file sizes.
 
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