IMAP (Internet Message Access Protocol) was created in 1986, but seems to suit the modern day world of omnipresent, always-on Internet connectivity quite well. The idea was keep users from having to be tied to a single email client, giving them the ability to read their emails as if they were in the cloud.
Compared to POP3, IMAP allows users to log into many different email clients or webmail interfaces and view the same emails, because the emails are kept on remote email servers until the user deletes them. In a world where we now check our email on web interfaces, email clients, and on mobile phones, IMAP has become extremely popular. It isn't without its problems, though.
Because IMAP stores emails on a remote mail server, you'll have a limited mailbox size depending on the settings provided by the email service. If you have huge numbers of emails you want to keep, you could run into problems sending and receiving mail when your box is full. Some users sidestep this problem by making local archived copies of emails using their email client, and then deleting them from the remote server.