When it comes to downloading messages, POP accounts will keep messages on the server by default. This means that messages will be still available when logging in through webmail or using another mail client. However, if space on the server is limited, Postbox can be configured to remove messages from the server when downloaded, or after a certain amount of time.
To configure if your POP account will leave messages on the server or not, go to:
Mac: Postbox | Preferences | Accounts | <your POP account>
Windows: Tools | Options | Accounts | <your POP account>
and check the corresponding checkboxes.
If you cannot find these settings, it could be that your account is setup as IMAP instead of POP.