http://auto.howstuffworks.com/rotary-engine.htm That link should clear up any issues with the understanding of how a Wankel rotary engine works. The wankel rotary engine has many unique advantages and disadvantages.
General Pros vs piston engines on equal fuel:
Fewer moving parts for less maintenance. Low weight. Lower vibration. Smaller size. Generally better fuel consumption at upper spectrum of the RPM range (due to lack of pumping losses @ higher rpms vs pistons). Wide applicable RPM range (some RX7 rotaries are operated daily within 2K-14K RPM). High Horsepower per engine displacement size due to longer combustion cycle (not a stroke as things move circularly).
Cons vs piston engines on equal fuel:
Higher cost of initial construction (economy of scale advantages far less than piston engines). Poor fuel consumption at lower RPM ranges. Poor engine response and power at low rpm ranges. Higher emissions in general. Possibility of "flooding" the engine during stops and restarts (a challenge to overcome in aviation).
I know a couple of people with experimental planes flying 20B mazda rotaries. Their greatest fear is the difficulty of inflight restart due to the flooding issue. This threat is greatest during cooler weather and engine temperatures. However, ensuring the proper warmup of the engine prior to departing is a part of the checklist.
Their secondary fear is lack of quickly available power during final approaches, however both of the pilots have special gearing in their props which hold the engine over 4000 rpm or so at landing speeds to keep the engine as close as possible to its 6000-8000 rpm main powerband.