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The carbon cycle is the biogeochemical cycle by which carbon is exchanged among the biosphere, geosphere, hydrosphere, and atmosphere of the Earth.  Carbon is present throughout the natural environment in a fixed amount. It takes many forms and moves through the environment via the carbon cycle.  The carbon cycle is the circulation and transformation of carbon back and forth between living things and the environment.
Both Photosynthesis and Cellular Respiration are complementary in that they produce products needed for each others processes. That is, the products of one process becomes the reactants of the other to complete a cycle of sharing carbon based organic chemicals.  Both processes are similar in that they move hydrogen ions charged atoms) around, have electron transport chains, and a series of enzymes. However they act in reverse to each other.

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Cellular Respiration is set of metabolic reactions and processes that take place in the cells of organisms to convert biochemical energy from nutrients into adenosine triphosphate (ATP), and then release waste products.
Cellular respiration is the process of oxidizing food molecules, like glucose, to carbon dioxide and water but whose purpose to produce the useful energy molecule ATP.
The chemical reaction (equation) can be basically shown like this:

C6H12O6 + 6O2 + 6H2O → 12H2O + 6 CO2
Sometimes the six water molecules are left of the of the chemical reaction (equation). The energy released is trapped in the form of ATP for use by all the energy-consuming activities of the cell.
The process occurs in two phases or stages:
The first stage breaks down a molecule of glucose (sugar) into two smaller molecules (pyruvate) and this is called: glycolysis. This happens in the cytoplasm using enzymes.
The second stage breaks these two molecules down further using oxygen into simpler molecules of carbon dioxide and water. This happens in the mitochondria. However other chemicals are indirectly made so that hydrogen atoms are freed up used to activate the enzyme ATPase to make the energy molecule ATP which is the main goal of respiration process in cells. Of course, only complex Eukaryotic cells have mitochondria..so respiration in simpler cells like bacteria happens a bit differently.
Plants have mitochondria for respiration but produce far more glucose (and the sugars and starches formed from it) than they can use as an energy source (but of course use it to builds their cells) and extra oxygen gas since they generally do not have the same energy or material needs for nutrients as animals etc.  Hence the names "producers" and "autotrophs" (make their own "food" themselves without needing to consume other life forms).

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EndosymbioticTheory states that energy organelles like mitochondria and chloroplasts may have been prokaryotes (bacteria) in the past. The evidence for this theory is that both organelles have their own set of genes (DNA), ribosomes (for making proteins), and reproduce by binary fission just like bacteria (prokaryotes). This these organelles are the first symbiotic relationships between life forms enabling more complex forms of life to evolve.  By L. Margulis and others.

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Photosynthesis is a process used by plants and other organisms to convert light energy into chemical energy that can later be released to fuel the organisms' activities (energy transformation).
It involves chloroplast membranes absorbing light energy with pigments like chlorophyll and converting it to free electrons through a series of proteins to power the process of connecting the reactants of carbon dioxide gas molecules with water molecules to produce glucose (a simple sugar-carbohydrate) and oxygen gas as products.
Basically it has two stages: light-dependent reactions-absorbing light energy, and light-independent or "dark" reactions which puts together carbon dioxide atoms and hydrogen atoms (from water) into glucose.
However it is, in detail, a complex series of reactions (photosystems I & II; electron transport chains; Calvin Cycle: and enzyme ATPase).  Oxygen gas form from the oxygen atoms from the splitting of water (H2O).

This process is common among plants, some protists like phyto-plankton, and cyanobacteria (which do not have chloroplasts but the entire cells acts like one).
Temperature, carbon dioxide concentration and light intensity and wavelength (color) are factors that can limit the rate of photosynthesis. Photosynthesis is most effectively powered by red and blue light or full visible spectrum of "white" light.
Thus, photosynthesis has changed the evolution of  planet by releasing oxygen into the atmosphere and giving life a turbo boost to the energy they can use to build more complex and multi-cellular bodies.