What is the difference between `mange` and `manges`?
In English one word can have many forms "I am going to eat", "I am eating", "I ate", "I have already eaten" depending on the context of the word, these are called conjugations.
Mange and Manges are two conjugations of the word manger "to eat". Throw it into google and you will find: http://french.about.com/od/verb_conjugations/a/manger.htm Which will tell you the context of the word. In french there are a lot more conjugations than English, but mostly they follow a regular pattern. I strongly suggest doing some investigation outside Duolingo to find out what the patterns are.
Oh, forgot to note that French has different conjugations depending on the subject. So while "I eat" and "you eat" use the same form in English, in French they are different "Je mange" and "Tu Manges" respectively.