Let's start with a quick review.
Photosynthesis is this process:
CO₂ + Water [+ light energy] --> Oxygen + sugar
Cellular respiration is this process:
Oxygen + sugar --> CO₂ + Water [+ energy]

Plants do both. Photosynthesis happens in the light. CO₂ in the surroundings is taken in. When there's no light, the plant still needs energy, so it does cellular respiration, releasing CO₂ to the area.

Photosynthesis happens only in cells that have pigment, such as green chlorophyll. The chlorophyll is in organelles in the cell called chloroplasts where part of the cycle takes place, but the part that uses CO₂ is in the main part of the cell. The CO₂ is taken in through holes in the leaf called stomata or stomates.

The sugar molecules made during photosynthesis travel throughout the plant. Cellular respiration happens in all cells because all of the cells need energy to stay alive. Most of the CO₂ is produced in organelles called mitochondria.

If we take a closer look at the molecules, here's what's happening:
6CO2 + 6H₂O [+ light energy] --> 6O₂ + C6H₁₂O₆

As you can see, every atom that comes in one side of the equation has to go out the other side, they are just rearranged in different molecules.

As you likely know, plants are able to manufacture their own food in a process we call photosynthesis. In photosynthesis, CO₂ from the air and H₂O from the soil react with the sun´s energy to form photosynthates (sugars, starches, carbohydrates, and proteins); O₂ is released as a byproduct. As you mention in periods of darkness there is no light energy which causes elevated CO₂ concentration in the plant as it requires light energy to break the CO₂ down into glucose and oxygen.

The actual gas exchange is complicated in plants. While plants have a very elaborate liquid transport system (water in through roots, etc.) this system does not participate in gas transport. Each living cell in plants is located close to the surface so that the distance gases must diffuse in even large plants is not great. Parenchyma cells in leaves, stems, and roots provide an interconnecting system of air spaces used for the diffusion of gases. In the plasma membrane of plants there are aquaporin cells which also aid in the diffusion on CO₂. In the leaves, O₂ and CO₂ are exchanged through pores called stomata. The stomata open when they are exposed to light and close when there is an absence of light. Guard cells surrounding the stomata regulate this process through osmotic pressure. An increase in osmotic pressure in the guard cells is caused by an uptake of potassium ions (K+). Blue light is absorbed by the phototropin which activates a proton pump in the plasma membrane of the guard cell (the pump is driven by a product of photosynthesis called ATP so light is required for this to happen). As protons are pumped out of the cell, the interior of the cell becomes increasingly negatively charged which attracts K+ ions into the cell, raising the osmotic pressure. When the K+ concentration in the guard cells exceeds that in the surrounding cells the stomata is forced open. In the absence of light, the pump stops, the K+ concentration drops, and the stomata closes. This helps regulate CO₂ concentration levels at night. So while CO₂ concentration is higher at night due to a lack of photosynthesis, the plant does not continue to absorb CO₂ throughout the night as the stomata are closed and gas exchange is not possible.

Interestingly, because CO₂ levels and the stomatal index are inversely related we can actually learn about past levels of CO₂ in the atmosphere from fossilized plant leaves.

Plants survive by converting sunlight to energy. However, since energy is always required by the plant and sunlight is not always available, plants funnel this energy through energy-storing carbohydrates (sugars).

At all times, plants break down carbohydrates such as starch and sugar into energy in much the same way that animals do, through the citric acid cycle. In this process, the carbohydrate bonds and oxygen from the atmosphere are carefully reacted to form CO₂, water, and ATP, a high energy molecule used to power nearly all cellular functions (gene transcription, translation, protein pumps, etc).

When the sunlight is available, the plant captures the energy from photons in temporarily charged molecules in the chlorophyll. These molecules are then used to power the reverse reaction described above, H₂O + CO₂ -> sugar + O₂.

Since the sugars produced by plants are used for much more than energy production, there is a positive net consumption of CO₂ and production of O₂.

The chemical reaction that plants use to create sugar produces oxygen as a byproduct and requires energy in the form of light, so when plants are exposed to light, they use it to make sugar. Sugar is extremely useful for plants because their bodies are mostly composed of the stuff (cellulose, the chief building material of plants, is being a polymer of sugar). However, plants can also use sugar in the same way we do: they burn it to release the energy stored within it, thus allowing them to survive periods of darkness when they have no other source of energy. Just as our burning sugar consumes oxygen and produces carbon dioxide, a plant's calling upon its sugar reserves and burning them does the same. The only reason why a plant, over its life, produces more oxygen than it consumes, and consumes more carbon dioxide than it produces, is because the sugar used to build the plant's body is never burned for energy during the plant's lifetime, and so all of the oxygen that went into those sugar molecules is stuck in the atmosphere.