Mail aplication

Which email account are you referring to is not clear from your post. Google, iCloud, AOL, Exchange, Outlook, Yahoo or any Other

For email, this is called Push / Fetch Settings. You may know that there are two settings (Fetch and Push Mails). What have you set? Watch the GIF below

Google, AOL, Yahoo - Only the Fetch mail Option is available

iCloud, Exchange, Outlook - Both Fetch and Push Mail option is available

1. Push Mail: Whenever possible the email client will push emails to all your devices.
2. Fetch Mail: You need to set How you want it to be fetched, Automatically, Manually, Hourly, Every 30 mins, every 15 mins
   1. Automatically: will fetch when the iPhone is powered and on WiFi
   2. Manually: with fetch when you open email, open Inbox and swipe down
   3. You get an email notification as soon as the client-server receives an email and your device is connected to the internet, irrespective of whether the mail is in your device's Inbox or not

This gif may assist you