When I change the mode, the mesh looks different. Edit mode-skinny, object mode-fat. Before I made the skeleton, everything was fine.
Make sure that before you do automatic weigthing, that you apply the scaling to both your mesh model, and you rigifie armature [CTRL + A]
once all scales are applied it shouldnt deform, i see this happen often with people who begin with this.
You applied an armature on your mesh. This effects the mesh only when you are not in edit mode. In edit mode you can see the original shape and in all others the deformed shape (the same when you apply other modifyers on your mesh). Your rigging was not very succesfull, as already mentioned. You have to apply the sam scale on the armature and the mesh and make sure, that they both have the same origin. Otherwise it looks a bit creepy after parenting them, like in your case.