What do you mean by “notifications” - messages that pop up on your screen to show messages that have arrived, or new emails downloaded to your iPhone?
The Apple Mail application is just a front end to your email providers; it has no control over the content that is sent. If you have an iCloud email address or a Microsoft Exchange email address you can receive “push” email, where new messages are sent automatically to your device. Gmail also has this feature, but only with the gmail app. If you have Notifications enabled for mail in Settings/Notifications/Main when a message arrives for these accounts it will create a notification using the settings you select.
For other email providers there is no push service available, because their email servers are not configured to push to Apple devices (actually to any devices). For these go to Settings/Apps/Mail/Mail Accounts/Fetch New Data and choose Fetch, which will periodically check for new email; choose how often at the bottom of the page.