It is possible to directly edit files in a repository stored on GitHub through the web interface. Most of us would never do this but it has its uses.
Since February 2016, you can add/create/upload new files from the GitHub web interface.
See "Upload files to your repositories ":
You can click the “Upload files” button in the toolbar at the top of the file tree.
Or, you can drag and drop files from your desktop onto the file tree.
This replaces the file creation described in the 2012 answer, and add the upload capability (for one or several files).