I'd really like to hear from you about the properties of your display
so that I know how to adjust images and what areas are still visible
and what not. I came up with this idea when I saw my photos on the
laptop of my sister. Her display can show way more black than
mine and so contrast was different and the photos looked different.

Because judging the contrast of a photo is always very subjective,
I've made two test images which I'd like to ask you to have a look
at and please tell me the lowest number you can see in the
white image and the lowest number you see in the black image
.

 

White range:

Black range:

 

As for me: white = 6; black = 14, barely the 13
This means that I can't see anything past 86% black and 95% white on my monitor.

UPDATE
After 20 responses the average white level is 4 and the average black level is 8.
Thus with my monitor's 6 - 14 I have a narrower band of brightness and will see
much more contrast in the image than the average viewer. You can use these
two images to open them in your favourite photo editor and adjust the curves or
levels on the white and black until you can view the 4 of white and the 8 of black.
You should then be able to see how it would look like on an average monitor, at
least approximately since you don't know about the contrast in the midtones.
For example based on this data I made following curve:
20 -> 33, 64 -> 76, 128 -> 128, 191 -> 172, 235 -> 215
If that works I'll use that curve in the future to adjust the contrast of a picture.