My problem is exactly as described in this review. It’s quite normal when you buy something new from a shop you’ll be relying on the shop to recommend accessories such as this Apple mini-DVI to DVI adapter for a external display. It was just $19, no big deal but it was completely useless. I would have expected the shop to warn me that a DVI-D equipped display is required for this adapter to be useful.
Learning new things along the way I found this good article about various kinds of connection for display.
Now if I really want to use dual monitor or larger external display I have to replace my very old FlexScan L350 by a modern LCD display that supports DVI-D input.
In Mac it’s called Clamshell mode if you are using external display with the MacBook lid-closed. This article shows you how. In my case I would like to run MacBook in dual monitor mode.