A HTTP Server with WebDav active is a server where you probably can update, delete, move, copy files. Sometimes you need to have valid credentials (usually check with HTTP Basic Authentication).
You should try to upload some webshell and execute it from the web server to take control over the server.
Usually, to connect a WebDav server you will need valid credentials: WebDav bruteforce(Basic Auth).
Other common configuration is to forbid uploading files with extensions that will be executed by the web server, you should check how to bypass this:
Upload files with executable extensions (maybe it's not forbidden).
Upload files without executable extensions (like .txt) and try to rename the file (move) with an executable extension.
Upload files without executable extensions (like .txt) and try to copy the file (move) with executable extension.
Davtest try to upload several files with different extensions and check if the extension is executed:
davtest [-auth user:password] -move -sendbd auto -url http://<IP>#Uplaod .txt files and try to move it to other extensions
davtest [-auth user:password] -sendbd auto -url http://<IP>#Try to upload every extension
This doesn't mean that .txt and .html extensions are being executed. This mean that you can access this files through the web.
You can use this tool to connect to the WebDav server and perform actions (like upload, move or delete) manually.
This vulnerability is very interesting. The WebDav does not allow to upload or rename files with the extension .asp. But you can bypass this adding at the end of the name ";.txt" and the file will be executed as if it were a .asp file (you could also use ".html" instead of ".txt" but DON'T forget the ";").
Then you can upload your shell as a ".txt" file and copy/move it to a ".asp;.txt" file. An accessing that file through the web server, it will be executed (cadaver will said that the move action didn't work, but it did).
If the Webdav was using an Apache server you should look at configured sites in Apache. Commonly: