All living organisms respire in some manner. Plants always respire but don't always photosynthesize. This is why plants give out oxygen during the day and carbon dioxide at night.
Plants require energy to maintain the function of their cells, normally obtained through photosynthesis. However, photosynthesis does not do all of the work required with respiration required to convert the necessary minerals into glucose. Photosynthesis occurs in the chloroplasts of plant cells where carbon dioxide (from respiration) and water (drawn through roots) convert energy from the sun and chlorophyll into a substrate of glucose and oxygen. This action proves respiration is necessary for plant cells.
Placing a plant in a room with varying levels of light will prove that while the energy needed to photosynthesize is absent, the act of respiration continues An example of this can be seen in non-evergreen plants shedding their leaves in winter, during which they respire while feeding on glucose deposits stored within.
The act of respiration is not the same as breathing. According to the Merriam-Webster dictionary, respiration is "any of various energy-yielding oxidative reactions in living matter." Plants respire in two manners with photosynthesis yielding oxygen and without photosynthesis yielding carbon dioxide.