From the location where you want to create a new directory, click the "Create new file" button. In the space for the filename, enter the directory name to create, and add a trailing slash. You will then be presented with a blank editor for your new file. Enter the filename at the top and start editing.
If you simply want to create a new directory, you'll need to create a placeholder file to put into it, and commit it. GitHub doesn't like empty directories.
Moving/Renaming a file
Edit the file.
Move to a subdirectory of the current location: prepend the filename with "/" and start typing the destination subdirectory name.
Move to a location "above" the current location: prepend the filename with "../" and start typing the destination directory path.