The reaction force when you place a cup on a table is also known as the normal force and is equal to the weight of the cup. By Newton's third law, every action force has an equal but opposite reaction force. The action force is the force of gravity on the cup which pushes the cup on the table. Thus its reaction force is the normal force from the table.
The gravitation force is the force that the Earth exerts on an object in order to pull it toward the center, which by definition is equal to the weight of the object. The normal force is a reactive force that a surface exerts to support an object that is placed on it. This normal force can be vertical, as occurs with the cup on a table, or horizontal, as occurs when a person pushes on a wall. It is not in all situations that the normal, reactive force from the table is equal to the weight. If an external force were applied to the cup in the direction of the table, as in someone's pushing down on the cup, then by Newton's third law the reaction force would be equal in magnitude to the weight of the cup plus the extra external force.