I would like to take my HDD with Debian Lenny installed on it and plug it into a totally different computer.
Does my Debian will handle the hardware changes? And if not, what do I have to do?
If this was the first drive on the old computer and you plug it in so that it's the first drive on the new computer, then as long as the hardware is compatible with your kernel, everything else should work just fine, except probably your network configuration (unless it's got the exact same network setup as well).
Otherwise, you'll probably need to create a bootable USB stick or CD (and test this on the new computer). When you move the drive to the new computer, if it fails to boot on its own, boot with the USB stick and: