HTML only
You could use something that (on drag and drop) automatically creates an invisible HTML form, a file input, copy the path of the filename into the fileinput and submit the form.
You can create the form inside an invisible iframe to send it in the background without changing the current page. You know, standard Ajax procedure.
A little help for dropping/pasting
I don't know if HTML allows dropping file items. If it doesn't you can look at the HTML 5 specification that Google is trying to make forward.
Another option is using some kind of rich client component (Java Applet with Swing or Flash, or Silverlight, or wathever) at least to grasp the dropping of the file (or the pasting) and creating the HTML form.
Why I prefer sending a form
I prefer the creating of the form over the applet sending the file because it doesn't require another special port at the server or something like that.