One of them I like to call bacteriorhodopsin photosynthesis. It is done by certain Archaea, notably Halobacteria like Halobacterium. Bacteriorhodopsin molecules pump hydrogen ions out of the cells with the energy that they get from incoming photons. The ions return through ATP-synthase complexes, thus completing this chemiosmotic energy cycle.
The other I like to call chlorophyll photosynthesis. The most familiar form of it occurs in the chloroplasts of plants and algae. Chlorophyll molecules energize electrons stripped from water molecules, electrons that are eventually transferred into various biological molecules, effectively adding hydrogen to them. Cyanobacteria do the same kind of photosynthesis, even down to a lot of details, and that is a big piece of evidence that chloroplasts originated from cyanobacteria by endosymbiosis.
There are other bacteria that do photosynthesis, various green and purple bacteria, but more limited forms, without splitting water. While cyanobacteria and chloroplasts have two kinds of antenna complexes, Photosystems I and II, these other bacteria have either I or II, though their electron transfer otherwise works similarly. Instead of water, these bacteria use such electron sources as Fe++, hydrogen, elemental sulfur, and sulfides.