Compared to a NASA space shuttle or China's Shenzhou spacecraft, a bottle rocket is a relatively simple affair -- just a soda bottle full of water and compressed air. But that simplicity is deceptive. A bottle rocket is actually a great way to understand and think about some basic concepts in physics, like different forms of energy, its force and potential.
An object has potential energy by virtue of its configuration or its position in a force field. If two positive charges move closer together, they have increased potential energy. If you take air and compress it, this inputs energy, and the increased pressure of the compressed air is a measure of its potential energy per volume. When the bottle rocket uncaps, the air inside has greater pressure than the outside air, so it expands and expels water from the bottle. For every action, there is an equal and opposite reaction; so downward force exerted by this expansion and expulsion in turn pushes the rocket upwards. The potential energy stored in the compressed air translates into kinetic energy.
Kinetic energy is the energy of motion. A moving or falling object like the bottle rocket has kinetic energy. Molecules and particles inside an object have kinetic energy as well, because they constantly vibrate or move. As gas molecules collide with the surface of the material confining them, they exert force on it. The force divided by the area is equal to the pressure. That's why reducing the volume of a gas increases its pressure -- the molecules are confined to a smaller area, but their average kinetic energy has not changed, so the force they exert on the material around them increases.
Gravitational Potential Energy
As your rocket rises, the kinetic energy of motion translates into gravitational potential energy. The rocket is moving further away from Earth's surface, so just like a negative and positive charge moved away from each other, the rocket has higher gravitational potential energy as it climbs farther from the ground. As gravity pulls on it, its velocity decreases until it reaches a point where all the kinetic energy has been transformed into gravitational potential energy. At this point, the rocket begins to fall.
Falling to Earth
As the bottle rocket falls, gravitational potential energy transforms into kinetic energy, and the speed of the bottle rocket rapidly increases. Eventually, it strikes the ground, where its kinetic energy dissipates as random motion of molecules in the pavement -- in other words, as heat.
You may notice that during the rise and fall of the bottle rocket, no energy "disappears" -- all energy either transforms from one form to another or changes from heat to friction and air resistance. The first law of thermodynamics holds that energy can neither be created nor destroyed; it merely changes from one form into another.