This is a very brief and basic definition.
A suffix tree is a binary tree which, when traversed from the root node (at the very top) to a leaf node (square, numbered), spells out a suffix of the word for which it is built. Suffix trees can be built in O(n) time, where n is the length of the string for which we are building it. The figure above shows the suffix tree for the word banana, where n = 6. (Never mind the $ for now!)
Why is it useful? Suffix trees are arguably the most used data structures in stringology, and have a vast multitude of uses. For example, if I want to find out whether the string ana occurs in banana, I can attempt to spell out each letter in the suffix tree. Starting at the root node, I successfully spell a (see leftmost solid edge on figure above). From this node, I then successfully spell n and a (which have been merged in to one edge, as the above figure represents a compressed suffix tree). An occurrence of ana can now be reported. This process requires O(m) time, where m is the length of the string I am looking for, in this case, ana.
For an in depth understanding of suffix trees, read chapter 5 of:
Gusfield, Dan. Algorithms on strings, trees and sequences: computer science and computational biology. Cambridge university press, 1997.