Well that is a hard one to answer as the list varies from person, to photographer, to photography school and publications. Figures I have seen myself vary from 6 to 50 different types. Then of course the there are categories with in each type.
I will endeavour to cover as many as I can here to try and give an insight and perhaps inspiration to you. If I miss any please feel free to let me know so I can include them.
From Aerial Photography to Vernacular Photography and everything inbetween and beyond - this could prove to be an interesting journey.