Is respiration the same thing as breathing?

No, they are not the same (although they do both involve the respiratory system!).

The key differences:Breathing is the process of moving air in and out of the lungs - like when you take a breath.Respiration is the reaction, carried out in cells, which produces energy.
Further information about breathing:
Air moves in through the mouth or nose, down the trachea (windpipe), into the lungs through the bronchi and bronchioles and into the alveoli. Gas exchange takes place here (oxygen enters the blood, and carbon dioxide leaves the blood). Then air is moved in reverse - from the alveoli, back through the bronchioles, bronchi, and trachea, and out through the mouth or nose. There are no equations to represent breathing, just the in and out movement of air.

Further information about respiration:
Respiration is happening all the time in body cells, and it takes place in the mitochondria. Most commonly this is aerobic (oxygen-using) respiration, which rearranges the atoms in glucose and oxygen to release energy, and produce carbon dioxide and water. This can be shown in a word equation:

                         Glucose + Oxygen -> Carbon dioxide + Water (+ energy released)

Or as a symbol equation:

C6H12O6 + 6O2 → 6CO2 + 6H2O (+ energy released)