You guys really seem to suck at explaining things very simple, and for what you can use them.
(Sorry for language, but I think this mathematical approach is not the way to do that...)
Look at a noise picture again:
What you can see there is, that the picture consists of grey colors from dark black to light white.
Each pixel in that image is actually only a value between 0 and 255.
But as you can see this picture does not consist of somehow mixed colors, it consists of regions which tend to be white and regions which tend to be black...
It has somehow a regularity, which makes them so awesome.
The very simplest answer to your question would be: You can generate mountains from them, for example.
And I really think this answer is pretty straight-forward and a good answer. See this image:
Edited this image to be one of our community ones
(You can also create other, not so smooth terrains
Here are some demo images in a post of one of our members: Terrain generator
The thing is: noise in the way you know it is used to create a big number of values, which look like they would create mountains. Think of each brightness in the image to be a number, which says how high a mountain should be at one point.
But noise is even more cool, you can use it to create explosion effects, lava textures, world maps, planet terrain (which is basicly the same as mountains, but more complex). Even minecraft uses noise to generate terrain.
Have fun exploring them and I hope this has helped you