I am not a scholar in the field of homeschool philosophies, so I am going to point you in the right direction rather than try to describe them all myself. This is by no means an exhaustive list, but does include some of the major philosophies.
Which do I practice in my homeschool? Why all four, of course. Seriously, though, I have taken aspects from each of these philosophies when coming up with my own. That is another beauty of homeschooling, developing a style that works for your family.