Push, if it works with your account, does it's job in the background. It does not matter whether the Mail app is open, in memory, or whatever.
Hold in mind that Push no longer works with Gmail in the Mail app. It used to work with the Exchange setup, but Google has done away with Exchange for personal accounts (though those with legacy setups can still use it, at least until they're forced to delete and re-add the account).
Fetch also works in the background (checks for emails at a set interval) and does not require the Mail app to be open or in memory.
Manual only downloads email when you open the app, or choose to refresh (by pulling down the email list) in the app; so it doesn't matter either.
To see what your account can do, and choose the setup you want, go t Settings > Mail, Contacts, Calendars > Fetch New Data > Advanced.
