Open Source Image Hosting Script: Chevereto
Chevereto is an open source image hosting script that can be used to create such a service with ease.
With a very chic interface, the script offers uploading of both local & remote files. And you can choose to resize the image too.
Chevereto is integrated with TinyURL too & an user can optionally shorten the URL of the hosted image while uploading it.
The script also comes with the documentation which covers creating a bookmarklet or a Firefox extension & more.
It is built with PHP & requires no database to run.