Google isn't using Flash, but actually some clever javascript (well, that IS what they're all about it seems :-) ). Using javascript and css, you can create a file chooser that lets the user select the file to upload. Then, you use a hidden iframe. The act of posting the form with your upload file targets the hidden iframe so that the result returned from the server on success goes into that hidden iframe. Using javascript, monitor the document body of that iframe to know when the file is uploaded.
This link appears to be a quick example of the basic concept: http://www.seemysites.net/projFolder/uploader/