IMAP is the ability to synchronize all your personal folders (not just the inbox) on an external client like Outlook, Thunderbird or Apple Mail. For a more complete definition of IMAP, I invite you to read this Wikipedia article.
Yahoo's implementation of IMAP is a pretty weird one. AT the time of the writing Yahoo! only "support" IMAP on mobile devices. This, however does not mean that IMAP is restricted to those smartphones. At first Apple announced that Yahoo! would offer free IMAP to anyone on the iPhone. Then Yahoo! released the protocole on BlackBerry and Windows Phone. IMAP was made official by Yahoo! on smartphone which means that there is someone in the mail team dedicated to make it work and to check on any issue encountered by some users.
However, it was recently discovered by some genius in the Yahoo! Mail Group that IMAP was actually wide opened to anyone...
...should you have a paid or a free account
...should you set it up on a mobile device or on your computer
...should you be using a 3G or a WiFi connection
So bring on the setting!
Now, one problem lies in the push system which means that you receive near-to-live updates as soon a new email message has arrived in your mailbox. The push notification works fine when you set up Yahoo! Mail BUT it only takes into account new email messages that have arrived in the inbox.
Also note that using IMAP on a desktop client IS NOT recommended by Yahoo! nor supported by the Mail team. This would explain the lack of push notifications and shows that the implementation of the protocole has not been fully made.