I can tell you about a couple of them.
Square taper is the "old school" type where the spindle is a tapered square shape and the crank arms are press fit onto the taper with a bolt. Works very well, but you need an extractor tool to get the crank arm off and many people ****er the threads on the crank arm, making removal difficult, but not impossible.
Cartridge BB is sealed bearing assembly that threads into the existing BB that had ball bearings. They last a long time and have no maintenance.
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Il faut de l'audace, encore de l'audace, toujours de l'audace
1980 3Rensho-- 1975 Raleigh Sprite 3spd
1990s Raleigh M20 MTB--2007 Windsor Hour (track)
1988 Ducati 750 F1