I have following elevation data of a region (the higher the altitude, the brigher the points):
How can I find out, which areas are filled by water, and which not?
Can I assume that if the elevation is below X meters, the place is filled with water? If yes, what is X equal to?
Your main question title indicates you're looking at sea level and coastal areas. But we'll start at the bottom of your question and work backward.
Consider lakes and rivers. Many of them may be significantly above sea level, even their bottoms, thus relying on elevation alone in the DEM cannot tell you if an area is water or not. Plus in a DEM large water surfaces generally appear as flat at the surface elevation. Expanding on this, and what iant alludes to, there are many places with inland areas that are below sea level (or theoretical zero in your DEM) that are not covered by water.
As Vince says, scale and resolution of the DEM are also a factor and basically you need ancillary data - elevation alone can't tell you. Imagery, maps, and vector data of coastlines are possible sources. Hydrology tools could also be potentially used in that they can fill 'up' and account for some of those isolated low areas not being covered until a certain elevation is reached at which point it spills in from surrounding areas. If you look up examples of studying sea level rise you'll find numerous examples of how just going up or down from zero doesn't quite work.
