I've tried to import an animated FBX model in blender, but as result I've obtained just the model itself with its skeleton but without animation. How am I supposed to import the model so that I can obtain the animation as well?
Blender FBX importing of animation is rather limited - only handles baked animation (that is, animated curves that are a bunch of points with linear interpolation between them), for transform (loc/rot/scale of objects and bones) and shape keys influences.
Beyond that, not sure which version of the FBX addon you are using, but FBX importer often finds many 'animation stacks' compatible with an object in FBX file, it converts them as Blender actions, and (at best) randomly affect one of those actions to the object. So suggest you check the Action Editor, and re-assign the desired action to your object.