Jonathan,
Can you please provide more details of your question? By "modifying the file", this could mean too many possibilities. To talk about security and permissions, you have to determine in what way your files are accessed first. By the way you asked your question, I am going to make these assumptions:
(A) You do not own your server.
You cannot make system-wide configuration changes to your server.
(B) Other people can login to use this server.
(This means having access to a command prompt of this server.)
You do not trust these people and do not want them to modify your files.
(C) Your web server is either Apache or Microsoft IIS.
.
By "modifying the file", there are some possibilities:
(1) When you visit the URL, it shows the original content of the file and a "Modify" button.
If you click the "Modify" button, the original content will be shown in a text area and a "Save" button comes out.
After you finish editing, you click "Save" to save the changes.
(2) There is NO editing interface.
The interface is like a "file manager".
You just upload another file with the same name to overwrite / replace the original file.
.
If what you want is (1) above, you have to write a server-side script (eg. PHP, ASP, etc.) (Of course you can use server-side scripts written by other people too!). Make sure your server does support the language (+ dependent libraries) you choose.
If what you want is (2) above, there are many ways to achieve, but I'm afraid a server configuration change must be done:
(1) You can add an FTP server to your server (if you can). Then using Firefox with the FireFTP plugin (or using Internet Explorer, if that is your choice), you can upload / download the file.
(2) You can add / enable the WebDAV module to your web server. Adding "WebDAV" function allows files to be opened / uploaded via HTTP. Microsoft call this "web folder". Linux and Apple call this "WebDAV" (the original name). There shouldn't be a problem to upload your files with Windows Explorer (Windows) / nautilus (Linux - Gnome) / Finder (Max OS X), even they are not browsers.
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Please provide more information to your question.
Kenji