The chemistry of rocket fuel is a complex field. There are many different formulations, some secret, some proprietary and their performance under even seemingly identical conditions can vary greatly depending on tiny differences unnoticed by the attending engineers. However, there are largely two separate fields of rocket fuels, and the process by which they work, on the molecular level, is the same. Producing thrust - the whole point of a rocket fuel - requires the rapid production of a large quantity of gas, which expels violently from the rocket nozzle, propelling the rocket along its trajectory. This large quantity of gas has to come from a relatively small amount of fuel, as there is only so much room inside a rockets interior.
Solid-state rocket fuels are exactly what they sound like: solids powders pressed together, often into pellets, and set off by either shock or an ignition source. Old fashioned black powder is a classic example, containing carbon (which acts as the actual fuel), saltpeter (aka potassium nitrate, which acts as the oxidizing agent - the key to any rocket fuel; a substance which can strip electrons from other atoms, consuming and transforming them) and sulphur (added to boost performance). Upon ignition, the oxidation agent consumes the fuel and the result is a rapid release of a large amount of gas: the rocket boost, the trail of smoke you see behind rockets in flight.
Other solid-state fuels include potassium nitrate mixed with various forms of sugars, with the same concept at play: the saltpeter consumes the sugar, releasing large amounts of high temperature gases which escape through the rockets nozzle, providing thrust. Metals can also be used as fuels; powdered zinc and aluminum often find use in solid-state thrusters.
Liquid-fueled rockets play off the same principles. Here, the components aren't solids, they're in liquid form, and are mixed at the moment of ignition. Examples of oxidizers used included the highly toxic nitrogen tetroxide and also liquid oxygen. The source of fuel is most commonly liquid hydrogen. Liquid oxygen combining with liquid hydrogen produces a seemingly benign product (simple water) but it does so with such vigor and incredible force that the temperatures and thrust involved are almost inconceivable. This combination finds use in our space program and, by logical extension, many applications where such incredible thrust is needed, and the cost of handling such troublesome and volatile fuels such as liquid oxygen are not a concern.
Our nation has spent decades refining and perfecting the science and design of rocket fuels, but at the core of the subject, it's a simple matter: an oxidizing agent consuming some sort of fuel, producing gas which is then used for propulsion.