...similar to those produced by email clients like thunderbird or outlook, sliding up or fading in from the tray.
What are your notifications for? Is it for a program that already exists, or are you writing a new application?
I ask because Adobe AIR has some functionality for doing this sort of thing, either as a Flex based application, or an AIR application written in HTML/JS. But, you wouldn't use AIR unless this is something you were building from the ground up.
The simple popup: Look at http://jtoaster.sourceforge.net/
For the fancy stuff look at what Java 2D can do:
http://java.sun.com/products/java-media/2D/samples/java2demo/Java2Demo.html
Check Composite -> Composite FadeAnim.
Swing's Timer class (http://java.sun.com/docs/books/tutorial/uiswing/misc/timer.html) can be used for rudimentary animation - just use the timer to update the location of the panel containing your content.
I attended a JavaOne session last year where they did some pretty slick stuff with this.
Slides : http://progx.org/users/Gfx/FilthierRicherClientier.pdf
pay special attention to page 29 and on.
And here's a website from the same folks, but with more code and screenshots, etc...
The end result of all of this was very, very slick.