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).
Note that since early December 2012, you can create new files directly from GitHub:
ProTip™: You can pre-fill the filename field using just the URL.
Typing?filename=yournewfile.txt
at the end of the URL will pre-fill the filename field with the nameyournewfile.txt
.
I think using gist.github.com is the easiest option. Each gist is a Git repository that can be accessed from the web browser or from the shell, there's a comment button, and although gists are usually single files, they can hold multiple files.
The drawbacks to gists are obviously the cryptic URLs and the fact that you have to copy-paste file contents into the browser unless you use a separate tool.
If you just need to have them edit the same files (assumedly through the web interface), you could create a skeleton repository with empty or basic files, and then have the students fork that.
If you start a new project without a README file, it won't show.
You need at least one file in it. As it's shown in previous answers, you can drag your files or select it using finder or file explorer.