Until now, I have used Roundcube Webmail as my online mail client, but recently I was exposed to the Outlook 2013 Web App, and would like to replace Roundcube with it. The two problems that I am having are that I cannot find where to get the web app and all research that I have done says something about needing a Microsoft Exchange Server. What the heck is that? I am using a free web host, so I imagine that this could be a problem.
Microsoft Exchange Server is designed for large organizations that need to host their own mail servers.
The Outlook Web App (formerly "Outlook Web Access") was designed for corporate users to access their email without having to install Outlook on their desktops. The OWA website would be hosted on one of (possibly many) Exchange servers. Of course, all the servers would have to be the appropriate version of Microsoft Windows Server.
You appear to be looking for a way to check your IMAP mail. Microsoft Exchange Server is way too complicated and costly for what you're attempting to do.
Outlook.com (formerly Hotmail) can check IMAP mailboxes, is web-based, and is free with advertising. However, you would be storing copies of the mail on Microsoft's servers, and I'm not sure if it provides you with the features you're looking for.