I'm building a web app that relies quite heavily on email notification/responses. I obviously don't want to rebuild gmail from scratch(!) but I'm not sure how to integrate an existing webmail client into my site. I've looked at RoundCube, AtMail and Squirrel but still am not sure if they're what I need. For users of my app to use those applications would they be redirected away from my site, or can I use APIs to keep everything within my app? Related to that, I only really need users to be able to read and respond to messages generated by my application. Although a full email client might be useful for some, it's not essential. I wonder whether a private message system (which could, if needed, send out emails to users registered address) might be more appropriate. Is this something I could relatively easily put together myself, or should I again be looking for an existing solution?
I don't think, that integrating something like RoundCube to your site is a good idea in your situation, because those ones are meant to be stand-alone services, not a part of other ones, and do not provide API that can be used by your app.
Server side languages usually have a set of functions to work with IMAP protocol.
PHP: http://php.net/manual/en/book.imap.php
Python: http://docs.python.org/3.1/library/imaplib.html
You can find libraries for other languages too, I'm not sure what language you need, since you didn't specify it.
Imap usage requires some server configuration, but makes rather easy to fetch and read received emails from your script. Depending on your needs, you can send mails and reed and analyze replies.
Hope this information will be helpful to you.