I remember a very long time ago when I was first getting into software I was curious about how compression works. I started looking through some Wikipedia articles on compression and was blown away by the complexity, and concluded that you need to be a genius to even begin to understand how compression works.
But if you look past some of the legitimately complicated optimizations, and intimidating foreign names (e.g. "Lempel-Ziv"), some of the principles of compression are surprisingly simple and easy to understand.
Take for example the LZ77 algorithm:
This intuitive and simple 4-step algorithm is together with Huffman coding (the steps of which are also easy to learn) the foundation of the ubiquitous zip compression format.
I think it would be a good exercise for anyone learning programming to write a program that compresses and decompresses files using the LZ77 method.