Answer:
Photosynthesis and metabolism are among the most complex areas in biology so given the nature of this forum I've kept the answers simple and brief.
Carbon is of central importance to all biological systems due to its special bonding properties allowing it to form various bonds with other atoms and produce a wonderfully complex range of molecules used by life.
In photosynthesis inorganic carbon in carbon dioxide gas is fixed to hydrogen to produce sugar, an organic molecule. In this case the carbon gains electrons so it is 'reduced' and this process requires energy in the form of light. Once in sugar form, the process can be reversed and the carbon can be oxidised back into carbon dioxide during cellular respiration, releasing energy.
So in photosynthesis, the carbon from carbon dioxide is reduced to form a sugar molecule. When transitioning to respiration, the carbon in the sugar is oxidised to form carbon dioxide again in the reverse reaction to photosynthesis.
The carbon is transferred between molecules through various intermediate steps during these processes, involving enzymes (biological catalysts) to assist in cleaving specific bonds at each stage. During cellular respiration (an energy release reaction) as the carbon is successively oxidised electrons are liberated that are used as part of the energy release. These electrons are captured or 'carried' by special organic molecules called NAD and FAD (reducing them) which in turn can then be oxidised to produce the universal energy currency of life: ATP molecules. ATP is a small bio molecule containing a high energy phosphorous bond that can be broken to release energy to do cellular work. It is used by all life that we know of and is the ultimate product of cellular respiration.
