What kind of account? POP or IMAP. Sounds like IMAP since that is server based and you just remotely view messages with your computer or other devices. With IMAP when you delete a message it is removed from the server and that is the only place it existed. It you want to keep copies of IMAP messages locally on your computer you need to make a folder or folders under Local Folders and move the messages there before deleting them. It might be a good idea to learn about the difference between POP and IMAP and decide what works best for you.