I've got several layers in "boxes" with text in it, that means that the text MUST fit into these boxes.

I've found out that you can set the font-size with Cascading
Style Sheets that it can't be changed by the browser
(In IE you can choose if you want bigger or smaller font
size).

But now a tester told me that if you choose bigger fonts
in MS Windows (display properties, where you set up your
graphics adapter) the font-size in CSS is BIGGER while
"normal" HTML is not changing in size.

Is there a possibility to find out if a user got bigger
fonts activated or is there another "trick" so that the
font size is (mostly) ALWAYS the same (while not using
images) ?

The most effective way that I'm aware of is specifying the fonts in pixels -- which will pretty much display universally one way. For example, if you want "Hello and welcome to my site" to be displayed in the same size, use something like <span style="font-size : 11px">Hello and welcome to my site</span>. That should fix the problem.

Happy coding!