Snoopy's plane is a Sopwith Camel, which is a single-seated biplane fighter flown by the British during World War One. His enemy during these fantasies, known as the Red Baron, was an actual German pilot named Manfred von Richtofen, who was credited with 80 air combat victories during the war.
The Sopwith Camel was credited with shooting down 1,294 enemy airplanes during World War One, more than any other Allied plane. While difficult to handle, it was highly maneuverable when manned by a capable pilot. Twin machine guns were fixed directly in front of the pilot, and a synchronization gear had to be used to assure that firing the guns did not shear off the blades of the front-mounted propeller.