We strongly discourage the use of POP (also referred to as POP3) and always recommend the use of IMAP instead. POP was designed to download your messages to the place you were reading mail, every time you read it. (Using the "leave on server" option is basically just a hack.) It was introduced in an era when you were unlikely to have a computer at both home and work, much less a mobile phone that could read email, and webmail that gave you access to your email from any computer.
IMAP was always designed to store your messages on the server. You can tell your local mail program to cache a copy of everything, but a copy is on the server, too. Why do you want your email "in the cloud"?
If you read your email in more than one place (like webmail and an email program, or your desktop and your cell phone, or your work computer and your home computer), then IMAP is an absolute must-have. POP has a "leave on server" option, which is designed to mimic IMAP, so you can read webmail. But this is a hack, and sometimes it means you end up with duplicate copies in your mailbox, because something loses track of whether a message has already been downloaded. IMAP's initial design was for messages to stay on the server, and be read from multiple places. Other message data, like flags, marking a message as read or unread, creating new folders, or moving a message into a sub-folder, will happen across all your machines.
If you've ever had a computer crash, you'll immediately see the benefit of using IMAP. See, when you store your mail on our servers (or, hopefully, any ISP's), we're using computers that are designed to prevent failure and data loss. The mailbox you're using is running on redundant hardware, so that if one part of it fails, we can fix it without losing anything, and we're making backups, too. Your home computer has a single hard drive, and maybe you're making backups (you should), but maybe you're not. If it fails, and you're using POP, there goes all your mail. Data recovery is really expensive, and, well, most people just cry and say goodbye to their email. If you're using IMAP, though, just go get that hard drive replaced, and when you come back and set up your email program again, you'll see your mailbox refilling with all your old mail. Technology is a beautiful thing sometimes.
Finally, IMAP stores all your folders on the server. POP is for your Inbox only, even if you tell it to leave your mail on the server.
(Adapted from a post on the Pobox Blog - Modern Times: Make Mine IMAP!)