A saprobe is an organism that derives its nutrition from the dead remains of other organisms; a scavenger, if you will. Saprobicfungi usually live on dead vegetable matter (sticks, leaves, logs...), as they are the only multi-celled organisms that can digest the structural proteins cellulose and lignin, the two major components of wood (and, in fact, the two major components of plants’ cell walls in general). They aren't the dominant decayers of animal flesh, as quicker-moving (and quicker-feeding) things usually get there ahead of them. But they are the dominant saprobes of the animal structural proteins keratin (hair, hooves, fingernails) and chitin (shellfish shells, insect exoskeletons). Basically, fungi handle the really tough organic molecules that nobody else wants to bother with. (The picture on the right shows Onygena equina growing on a fragment of horse's hoof)
In older literature, you will see the term saprophyte used instead of saprobe. That term is now out of fashion, as its etymology carries some implication that fungi are plants. It also leaves the term saprophyte out in the cold, as plants can’t break down organic matter on their own; so there no longer are any saprophytes...