I want to create a website that offers users modules/widgets that can be dragged around, closed, minimized... like iGoogle (not necessarily looking like that, it's just an example). These modules/widgets should have the ability to dynamically load the contents from DB and show them. What tool/resource you suggest to do this as fast as possible. I know programming but I'm new to web and PHP and I'm supposed to deliver this project in 20 days tops! So I prefer the minimum amount of learning if possible.
Here's a tutorial from nettuts that teaches just what you're asking for (except the loading from DB part). The live demo is here
This isn't necessarily PHP, what you're referring to is client side. Take a look at jQuery, that allows the minimize/maximize, drag/drop, etc.
There is also other JavaScript libraries out there like MooTools and Prototype.
For this type of project you will require client side scripting i.e. Javascript (PHP is server side scripting)
JQuery makes writing Javascript considerably less tedious and JQuery UI allows you to add interactions to DOM objects such as: Draggable, Droppable, Resizable, Selectable, Sortable which will make creating the widgets much easier. http://jqueryui.com/
You might not need to write the windowing yourself the Elgg social network PHP framework has a theme called the Deyan Shell which pretty much does what you need out of the box: http://community.elgg.org/pg/plugins/project/752334/developer/vabel/deyan-shell