WebDAV (shorthand for “Web-based Distributed Authoring and Versioning”) is an extension of the standard HTTP protocol designed to make the web into a read/write medium, instead of the basically read-only medium that exists today. The theory is that directories and files can be shared—as both readable and writable objects—over the web. RFCs 2518 and 3253 describe the WebDAV/DeltaV extensions to HTTP, and are available (along with a lot of other useful information) at http://www.webdav.org/.
A number of operating system file browsers are already able to mount networked directories using WebDAV. On Win32, the Windows Explorer can browse what it calls Web Folders (which are just WebDAV-ready network locations) as if they were regular shared folders. Mac OS X also has this capability, as do the Nautilus and Konqueror browsers (under GNOME and KDE, respectively).
How does all of this apply to Subversion? The mod_dav_svn Apache module uses HTTP, extended by WebDAV and DeltaV, as one of its network protocols. Subversion uses mod_dav_svn to map between Subversion's versioning concepts and those of RFCs 2518 and 3253.
For a more thorough discussion of WebDAV, how it works, and how Subversion uses it, see Appendix B, WebDAV and Autoversioning. Among other things, that appendix discusses the degree to which Subversion adheres to the generic WebDAV specification, and how that affects interoperability with generic WebDAV clients.