Why Emails remain on my iPhone after deleting from desktop

It depends on the mail agent you use and how you set it up. With most modern email systems, they are IMAP (Internet Message Access Protocol) and when you set up an account as such, mail deleted in one place, like your computer will delete in all places. But if your mail agent is POP, they don't sync across devices.

Since we don't know what email you have or how you set it up, you might want to check with your email provider to see how to make sure you've set it up as IMAP as opposed to POP.