Atkinson cycle engine is considered to be an improved and modified version of the Otto cycle engine. It was invented by James Atkinson in 1882. Much like the Otto cycle engine, this engine has four strokes: intake, compression, combustion and exhaust. The duration of the strokes is different because of altered valve timings. The fundamental difference that an Atkinson cycle engine has is that during compression after intake, the intake valve is left open for a little while longer even as the piston moves up. Whereas, normally if the piston is moving up during compression, the intake valve is propelled to shut and seal preventing air-fuel mixture from escaping. Now, the escaping air-fuel mixture is beneficial in terms of efficiency. As compared to other engine cycles, this is beneficial in a unique way. The likely assumption is that if the air-fuel mixture seeps out with the rising piston, efficiency should decrease. However, the idea behind allowing the intake valve to remain open is more complex.
If you consider the compression ratio in an Atkinson cycle engine, it is less than your expansion ratio and that improves efficiency. Having said that, this efficiency does come at a cost and the cost is power or torque. In an Otto cycle engine, there is more air-fuel mixture in the chamber during compression as it is prevented from escaping. This creates a disparity in pressures in respective engine types. You enable the piston to produce more energy using less fuel.
It is noteworthy that the air-fuel mixture that is being released during compression in an Atkinson cycle engine is actually being propelled into another cylinder so it can be argued that the mechanism is not as inefficient as it seems. It utilizes the air-fuel mixture that was released earlier. The power stroke in this engine is slightly longer due to the high expansion ratio. At the same time, the exhaust stroke is shorter. The Atkinson cycle engine is befitting for hybrid vehicles as the cycle loses torque primarily due to a reduced compression ratio. Hybrid vehicles are equipped with an electric motor which pairs up with this particular engine to make use of high power that is generated at a lower speed. For Toyota, the Atkinson cycle engine proves useful in hybrids as this pairing of the electric motor allows an independent source to shut the engine down whenever possible. Many would argue that that is indeed the most reliable way of improving fuel efficiency. That is precisely why Toyota has given preference to this type of engine in hybrid electric models.
