They don't "store" it, it just floats in the cytoplasm until used. It "energizes" protein molecules, i.e. causes them to change shape.
All cells require the use of ATP. All cells need energy to function and ATP is one of the best source of energy in a cell.
When the cell reactions add another phosphate to ADP, making it ATP, that third bond is highly unstable. This means that phosphate bond stores a lot of chemical energy. When the phosphate is...
Mitochondria.
Most are created by the mitochondria through cellular respiration, but some amounts of ATP are also created by the cytoplasm via glycolysis.