I love the hover effects of StackOverflow.
I want to use similar feature
The new jQuery UI tooltip is quite good: http://jqueryui.com/tooltip/#custom-style
The animation is controllable, the tooltip is styleable, you can pretty much mimic the StackOverflow like tooltips.
Note that i linked you the "custom style" example. It shows you how to make custom styles for the tooltip.
Also that by default the tooltip shows the title attribute of an element. There is an example: http://jqueryui.com/tooltip/#custom-content where custom content is set in different ways to show you a few options.
I'm not sure what feature is missing.
EDIT: the custom content demo is bugged on the example site, check it in full page view: http://view.jqueryui.com/menubar/demos/tooltip/custom-content.html
EDIT2: New Solution
Looking at Anton's answer I thought that this could be done correctly by adding the tooltip's div inside the original div, and introducing hoverintent. Also a little sanitization of the code was in order. Check: http://jsbin.com/anegip/2/edit
http://jsbin.com/anegip/3/edit
With a little design, it's what you want.