That is a trick question. Lets break this into two parts so you can see why and hopefully get you setup the way you want.
When you setup an Email account there are two types. Push and Pull. So if you setup a Hotmail account, that would be Push. If you setup a Gmail account, it can be either. If you setup a POP3 account it will always be Pull.
If it is Push account, then you can get instant notifications. If it is setup as a Pull account, then you have to setup a frequency for iOS to check. Anywhere from 15 minutes to never.
But once you resolve that issue, then you need to setup the second side of the question.
Settings -> Notification -> Mail.
Once in there you can setup each account (I have 6 personally) to display the type of alert you want. So pick yours and then, you want to check out each setting to read its effect. Personally I setup my accounts with:
Notification Center: On
Alert Style: Banners
Badge App Icon: On
New Mail Sound: {What ever you want}
Show Preview: On
View in Lock Screen: On
Hopefully you can resolve your issue now.