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In brief, both photosynthesis and cellular respiration are processes of converting a source of energy into chemical or biochemical energy which can be used by an organism to fuel certain activities.
Photosynthesis is the process by which plants (and other photosynthetic organisms such as algae) convert carbon dioxide and water into sugars (such as glucose) and oxygen as a by-product using light energy from the sun. Light energy from the sun provides the energy to use ATP in order to build sugar. The sugar is used to store chemical energy, which can be used to fuel the plants activities. This occurs in the chloroplasts of plants.
Cellular Respiration, on the other hand, is a set of catabolic reactions that break down carbohydrates and sugars in order to use the energy stored in them, usually in the form of adenosine triphosphate (ATP) the energy currency of the cell. In essence, it is the reverse reaction of photosynthesis. Whereas in photosynthesis carbon dioxide reacts with water as catalyzed by sunlight to form sugar and oxygen, cellular respiration uses oxygen and breaks down the sugar to form carbon dioxide and water accompanied by the release of heat, and production of ATP. This occurs in the mitochondria, the powerhouse of the cell.
The relationship between the two processes are apparent in that photosynthesis captures light energy and stores it in a form that can be utilized by the cell through respiration - that is, in the form of a carbohydrate or a sugar. It can therefore be seen as a cycle by which light energy is converted to chemical energy, which can be used by the cell to fuel all other processes. A curious thing is that photosynthesis uses ATP to make glucose, only for respiration to use this glucose to make more ATP. There are a lot of possible reasons for this. First, there is not source of sunlight at night (or during times when there's not much sunlight like winter). Hence, instead of simply using sunlight to acquire ATP and use it to directly power cellular processes, it is better to store the energy in the form of sugars which can then be utilized as needed despite the absence of light energy. Sugars are also building blocks of various biochemical molecules needed by plants.
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