"regular IC" you're referring to is really the Otto cycle, named after the inventor, which is the standard 4-cycle engine used in nearly all cars.
Atkinson was another engine inventor who developed a slightly different engine, to get around Otto's patents. Main differences are that his piston-crankshaft linkage was more complex, using a pivot and a second link, and that he controlled the engine's power output by adjusting the intake valve timing rather than using a throttle valve to restrict the intake charge. This different method of power control ended up being more efficient because it reduced pumping losses (sucking through a closed throttle valve) at low rpms and loads, where most driving is done. The prius uses this intake valve timing method, and has a slightly offset crankshaft to reduce piston-cylinder frictional losses, but just uses one con rod, not the multi-link setup that Atkinson originally used. It's still a 4-stroke suck-squeeze-bang-blow engine, just the suck happens a little differently, and the geometry is changed a little bit. Overall result is that it's more efficent in the power areas where a car engine is typically operated.
