The wall is typical old fashoned lath and plaster. Yes, the little small boards are probably attached to studs maybe on 2' centers. New drywall can be applied over the old provided you shim the places where the plaster is missing to level it up. You will need to locate the studs so you can attach the new drywall to them, especially at the joints.
Thanks for the pictures and We wish you the best with the project.