Difference between 何も、何でも、and 何か

何か - Something
何も（ない） - Nothing
何でも - Anything


何も is only used in a negative sentence and 何でも is used in a positive sentence.
Examples:
今日は何も食べた。I didn't eat anything today.
冷蔵庫に何もない。There is nothing in the refrigerator.（ない informal form of ありません. ある is the verb 'to exist', hence 'there is nothing'=何もない）
何もいない。I don't need anything.

何でも食べるよ！I'll eat anything/everything!
冷蔵庫に何でもありますよ。I have everything/many things in the fridge.
プルゼンとは何でもいいです。Any present will do. (Lit. As for presents, anything is good for me.)

Source:http://www.yesjapan.com/YJ6/question/1418/what_is_the_difference_between_nanimo_and_nandemo