Push - mail server sends signal to your device when there's a new message and delivers it to your device
Fetch - your device checks the server for new message. If there's one, it'll download it to your device
Manual - your device and mail server do nothing. If there's a new mail, you will only see it when you get into Mail app. You will see no new mail at all if it's set to Manual and you never touch Mail app to check for new mail.
The difference between manual and fetch is that with "Fetch" setting, your device wakes up every X minutes to check for new mail. You don't have to get into Mail app. It'll tell you when there's one or more new mail.