The Normal map just won't work on my mesh (it's created using metaballs) but if I try anything simpler, for example, a cube, everything seems to be working. Could it be because of my computer capacity or do you know what else could be the problem?
By default image textures are mapped with UV texture coordinates. This means that , even if you don't see any input node for the Vector socket of your Image texture node, you should assume that your UV map is being used for spreading the colors around.
I wild guess that you don't have UV unwrapped your mesh yet, so there are no proper UV island to work with for your shader setup.
The default cube, as many of the primitive objects, comes with a proper UV unwrapping by default: that's why you see the texture there.
My suggestion would be to urwrap your mesh or use another texture coordinate system, like Object's coordinate with Box projection method.