The most commonly used small model rocket engines are the black powder engines such as the one shown below. These are the "traditional" model rocket engines that have been in production since the 50's.
Black powder model rocket engines are made of a paper tube with a clay nozzle, a solid pellet of black powder propellant, a smoke/delay charge, and an ejection charge as shown in this figure.
Booster engines are similar but lack the Smoke/Delay and ejection charge.
A model rocket engine is ignited by inserting an igniter in the clay nozzle putting it in contact with the propellant. At launch, an electric current is driven through the igniter, causing it so explode, igniting the propellant.
When the engine is ignited, the propellant burns, ejecting high-pressure gas out of the nozzle and producing thrust in the opposite direction.
When the propellant is completely consumed, the smoke/timer charge burns producing a smoke trail. The timer charge performs two tasks. First, it provides a smoke trail to help you follow the flight. Second, it lets the rocket coast to its maximum height before activating the ejection charge.
When the smoke/timer charge is exhausted, the ejection charge fires, which pressurizes the rocket body and deploys a parachute or other recovery device.
When the burn reaches the ejection charge, a small explosion deploys the recovery system.