I don't understand what happens to my 3d model when I apply the mirror modifier. It seems that it adds a lot of vertices that I can't remove anymore. Anyway the modifier is able to mirror the eye socket correctly. Check the pictures attached and help me to understand how to fix the problem. Thanks.
The mirror modifier has a center. You need to place your model on the correct side of its origin. You may just create vertices on one side of the local origin. You only need model on one side ... the left or the right. There are more flexible alternatives than this but this is the good way to start.
In the image lets assume I am trying to make a sphere with just half a sphere. The blue sphere is almost correct. I wanted the separation to be a little obvious visually for purposes of explanation.
The next attempt is where the model has too much separation.
In the last model I have unfortunately placed vertices on both sides of the X axis. This may be what you did also.
You can either remove the mirror modifier. Or make a selection on the undesired vertices and grow the selection with Control+, then delete the vertices.
I wonder if you are seeing another mesh in the same place as the bird. I suggest you change the names of the mesh to something like Bird so as to explain the issue better.