The rotary engine was patented in 1929 by German engineer Felix Wankel, and it could not be more different to the piston engines that exist in most of the cars on the road today. A rotary doesn't have individual cylinders like a piston engine -- so when you have an inline four, a V6, a V8, that's the number of discrete cylinders a piston engine has. Instead, a rotary has a single combustion chamber per rotor, and the way it works is fascinating. If you watch the video above, you should be able to understand it pretty easily.
Rotary engines are fascinating because they're built around a Reuleaux triangle, a rounded-edged triangle that's a Venn diagram of three circles. The video is split into two parts: first, the four stages of any four-stroke engine's internal combustion process as it exists in a rotary chamber, and then a sped-up version of the rotary motion that gives you an idea of how it spins in the chamber. Combustion happens, rotor spins, crankshaft turns, car goes forward -- simple. If you want a point of comparison, here's how a reciprocating (piston) engine works.
(Also, if you don't look at the image up the top and laugh, you don't really like cars.)
Bonus fact for anyone that's scrolled this far: Panasonic actually has a robot vacuum cleaner using the Reauleaux shape. It's called the RULO.