Whether emails will be kept on the server or removed when downloaded locally depends on the incoming server configuration for your email client.
For IMAP email accounts – messages will always be left on the server until deleted.
If you are using POP3, you have the option to choose whether the emails should be left on the server or downloaded on the computer where the email client is working and then removed from the server. The exact place to change this configuration varies depending on the email client you are using. Here are a few examples:
For Thunderbird when using POP3 account – go to Tools -> Account Settings -> Server Settings for the account -> “Leave Messages on the server” option.
For Outlook – Under Internet E-mail Settings -> Advanced Tab -> “Leave a copy of the messages on the server”
If those options are checked the email client will download the messages locally but also keep them on the server (for a specified period or until deleted – this is defined in the 2 sub-options below for both clients). If the “Leave messages on server” option is not checked – the messages will be automatically deleted from the server when downloaded locally by the email client.