Atkinson's U.S. patent (number 367,496, for us patent-adoring nerds) is pretty straightforward: about a thousand words of text and a few helpful diagrams. Or you can just read this explanation, which is far wittier than any patent.
The most common combustion engine these days is a four-stroke Otto cycle engine, where a piston goes up and down inside a cylinder and a spark ignites a mixture of gas and air. Same goes for an Atkinson cycle engine, so here's a quick refresher of the process:
Intake stroke: Sucks air and fuel into the cylinder
Compression stroke: Squishes the mixture so when the spark goes off, it will explode -- big time
Power or expansion stroke: Uses the force created by the explosion to move the piston down the cylinder
Exhaust stroke: Pushes the nasty leftovers of the combustion process out of the cylinder
In an Otto cycle engine, this is done in two rotations of the crankshaft: intake/ignition, then power/exhaust. In the original Atkinson engine, the inventor added a couple of linkages so that all four strokes could be completed with a single rotation of the crankshaft.
That in itself would improve efficiency, but Atkinson had another realization: if the compression in the cylinder were lowered and the power stroke was longer than the intake stroke, the engine would work more efficiently. It would take less fuel to turn the engine, which turns the wheels and makes the car go.
Imagine, if you will, the cylinder and piston. On the intake stroke, the piston doesn't move all the way down the cylinder. The intake valve, where the air and fuel enter the cylinder, doesn't allow as much of the mixture into the cylinder. Less mixture requires less compression. The piston moves back up for the compression stroke, and at the top the mixture is ignited. Boom! The force sends the piston back down the shaft of the cylinder in the power stroke, this time all the way down to take advantage of every last bit of force generated by the combustion. Then the piston moves back up to get the junk out for the exhaust stroke. Ta da! Four strokes, less fuel!
Of course, clever reader that you are, you probably realized that less fuel and less compression mean less power. You are correct. Even though the piston is allowed to travel further down on the power stroke than it does on the intake stroke, it's not going to generate as much power as it does in an engine with higher compression and a richer gas mixture.
The other challenge with this engine is that it requires lots of extra parts, which makes it tricky to assemble, not to mention expensive. Poor Atkinson had to achieve all this efficiency with springs and vibrating links and a red-hot ignition tube, which sounds like an excellent name for a band. Modern engineers have a much easier time of it.