Go to Settings > Accounts, and select your email account. Then scroll down to "Mail days to sync". This may be set from 1 day to no limit. If you plan to sync to no limit, you may want to be on WiFi so you don't chew through your mobile data.
This will also depend on what email service you are using. Gmail via the iOS app will only support fetch, and the fastest fetch is 15 minutes. So if you're expecting an email, it may take up to 15 minutes before you receive it.
If you use the services iOS apps, the emails should be immediate.