What are the differences between types of mattresses?

I am in desperate need of a new mattress, the one I currently use is older than I am (22). I’ve been looking at the different types and wondering what qualities they have. Is a memory foam mattress better than a box spring? What type of mattress is good for bad backs? How much does the thickness matter? What the heck is a pillow top? What brands do you prefer?

A sleep deprived Jellie needs your help.