To summarize what has already been posted many times in this thread:
Push
Push email - that is, email that appears on the phone as soon as it reaches the server - requires the email provider to send push notifications. Currently, Apple's iCloud email and Microsoft Exchange are the only email providers that support Push. Google used to, and Yahoo used to, but neither of them support Push with Apple's email app anymore. You must use the gmail and yahoo app, respectively, to receive Push email from gmail or yahoo
Fetch
If your provider does not support Push you can set up Fetch on the account. It will check for email periodically; you set how frequently you want it to check.
Manual
Email will only appear when you open the mail app.
You must set which you want to use in the phone's Settings for each account.