IMAP is the abbreviation for Internet Mail Access Protocol. This protocol is used to manage email mailboxes and receive the emails they contain.
If you are using an email program, for example Microsoft Outlook or Apple Mail, to access your email account using the IMAP protocol, your emails remain on the server. The messages are only downloaded completely when the message is opened. All tasks (e.g. moving or deleting emails, creating folders) are performed directly on the server. This gives the advantage that you can synchronise your emails on multiple devices, and the inbox and other folders on all of the devices are identical. You have to set up your email mailboxes with the IMAP protocol on each device you would like for this to work.