I'm using Mozilla Firefox 4 and I would like to be able to receive notifications similar to Google Chrome's built-in notification system, like so:
Is there an addon like this or similar?
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this extension appears to do the job - its experimental, so it might either stop working in future, or rolled into mainline firefox in future.
Yes, you can try the Tab Notifier Extension for Firefox.
It provides desktop notifications really similar to Chrome’s ones (e.g. more windows are piled up vertically), as you can see in this video.
It appears you are using Mac OS X, and so you should note that this extension needs Growl to work on Mac, that it's not supported by the last versions.
This extension other than supporting website coded for Chrome-like Desktop Notifications (eg: GMail is supported) also optionally enables to receive notifications virtually for any website by watching for changes to its title.
Disclosure: the original author of this Answer is the Extension coder
Extension to add desktop notifications into firefox, the way it works with Chrome and Chromium Browser : works with gmail, gtalk (gmail chat), imo.im, and all other websites that use this technology.