The energy is stored in the phosphate bonds. Energy is released when the bond break and ATP turns to ADP.
In the bonds of the inorganic phosphates.
ATP is an adenine molecule bonded to 3 inorganic phosphate molecules, hence the name adenosine tri-phosphate. Each of the phosphate molecules are identically charged (-3), and really don't want to be attached to one another, kind of like three angry siblings sitting together in the back of the car.
Once one of the bonds breaks, it leaves very quickly, converting its potential energy into kinetic energy. Even after it releases the third phosphate group, there is still quite a bit of potential energy in the ADP (adenosine diphosphate) molecule that is left.
Continuing the sibling analogy, the kids have a little more room in the back of the car after the younger kid left, but they're still generally unhappy. If given the opportunity, they, too, will leave very quickly.