The big picture is that there's a few compression algorithms.

Run Length Encoding

Huffman Encoding

Lempel Ziv (LZ77)

Burrows Wheeler Transform

The interesting part of all these specific implementations is their own specialized way of extending or combining these algorithms for their own niche (or tradeoff point).

Bzip2 is for example, compresses really well because BWT is expensive, but clever - rzip extends the LZ part of bzip2 to look further across the file (instead of a few hundred kb).

Zlib itself has enough of these flags exposed out, so all Zlib isn't really the same - Z_FILTERED, Z_HUFFMAN_ONLY, Z_FIXED, Z_RLE etc. Look at something like Zopfli to see how they can be remixed, provided the tradeoffs of CPU change from the historic positions of Zlib.

Beside Huffman coding, there is also arithmetic/range coding and now is becoming widely used ANS coding - e.g. LZFSE (Apple), ZSTD (Facebook), VP10 (Google): http://encode.ru/threads/2078-List-of-Asymmetric-Numeral-Sys...

Almost every one as far as I know where based on the info-zip code by Mark Adler and Jean-loup Gailly. That was used as the base at the time for zip/unzip in linux, winzip, and gzip ultimately.

good video series straight from Google Developers https://www.youtube.com/watch?v=Eb7rzMxHyOk&list=PLOU2XLYxms...

He covers all that, at least in passing. It's a fairly comprehensive rundown.