OWA (Outlook Web Access) is Microsoft Outlook messaging and collaboration over HTTP (WebDAV) to a Microsoft Exchange Server. The server side protocol is based upon an incomplete wrapping of MAPI by WebDAV at the server. OWA will give you access to most, but not all, features that are otherwise available in Microsoft Outlook.
WebDAV is an extension to the HTTP protocol that a allows people to collaboratively maintain files on remote computers. A standard use of this is for HTTP based messaging and collaboration.
MAPI is the native protocol used by Microsoft Outlook to access a Microsoft Exchange server. MAPI is a Windows-only and proprietary messaging and collaboration platform and is the only platform that gives a client application full access to all Microsoft Exchange features.