One of the first devices to qualify as a rocket was a wooden bird. The writings of Aulus Gellius tell a story of a Greek named Archytas. Around the year 400 BC he stunned the townspeople by flying a pigeon made of wood. Escaping steam propelled the pigeon suspended on wires. The bird is an example of Newton's third law of motion- every action has an equal and opposite reaction.
About one hundred years later, a Greek named Hero invented a similar rocket-like device called an aeolian. It used steam to propel it. Hero put a sphere on top of a water kettle with two L-shaped tubes on opposite sides of the sphere. The tubes let gas escape. A fire below the kettle turned the water into steam, the steam gave the sphere a thrust, and it rotated the sphere.
Dr. Robert Hutchings Goddard constructed and successfully tested the first rocket using liquid fuel. In 1914, Goddard received two U.S. patents. One was for a rocket using liquid fuel. The other was for a two or three stage rocket using solid fuel.
See the related question for more information on Space Rockets.