An email application determines what happens to the email once it's been touched by that application. So if emails aren't being removed from the server once pulled, one of two things is most likely occurring.†
(1) If you're using POP3, there is a option to leave messages on the server. You may also be able to configure the duration that an email will stay on the server. If you're using multiple devices to check mail, you may want to keep email on the server for a period of time.
(2) If the account is set-up as IMAP, the default behavior is to retain mail on the server. IMAP works similar to standard webmail interfaces. You would have to set options for each folder if you're using IMAP.