A WebDAV module shipped with Gallery 2.2. It's documented in the Gallery codex. There's not currently a WebDAV module for Gallery 1.x. By providing a WebDAV interface to Gallery, the WebDAV module supports adding photos and albums to Gallery directly from your desktop computer, e.g. per drag and drop from MS Windows Explorer. The user documentation includes screen shots.
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WebDAV is a set of extensions to HTTP, beyond HTTP/1.1. It's consistent with HTTP in that it uses request and response messages, messages have headers and a body, response messages have a status. Consequently existing web servers can generally support WebDAV. WebDAV defines some extra request methods, message headers and XML message bodies. These extensions basically add metadata and locking to the HTTP protocol (however additional WebDAV specifications add further specializations like versioning, server side search and digital calendar access.) The fundamental goal of these extensions is to support distributed authoring and versioning. This is one of the original goals of the web: That it should be writable.
WebDAV's basic primitives (metadata and locking) are quite general and map well to file system operations. WebDAV is an open standard based on widely adopted technology: HTTP and XML. Consequently many WebDAV clients have been implemented. Most operating systems support mounting WebDAV resources like file systems: