Push: If your account supports it, then whenever your email server receives an email for you, it will notify your device so that it can be download. It’s called push, because the server pushes the email to you instead of waiting for you to look for it.
Fetch: Your device periodically checks the server to see if you have new emails, and downloads them. The time settings control how often new email is fetched; 15 min, 30 min, 1 hr, etc.
Manual: Email is checked and downloaded when you open your mail app, or when you force a refresh/check in your mail app. If you have both Push and Fetch turned off, then email is only checked with the app open.
Automatic: I’m not certain, but it is probably a variation of Fetch, where email is checked more or less frequently depending on whether you are on a local wi-fi network and/or the time of day.
In practice; if you need to know about new important emails throughout the day, then Push is the fastest. Next best is a short Fetch time. If you choose Manual, you won’t know you have new email until you open your mail app.