Boeing probably produces the most airplanes for the most commercial airlines that are out there. The United States mainly uses Boeing, but many commercial airlines overseas use the Airbus. Airbus has the biggest plane at Airbus A380. It has a double deck and has four engines. Many passengers take both the Boeing and Airbus planes that are bought by the commercial airlines.
It seems that Boeing has been in existence longer than Airbus. Since Boeing has been around much longer, they have been able to learn more about plane over the time they have been in operation. Airbus has been in operation significantly as early as 1970, but mostly in the year 2000. They compete with Lockheed and Boeing for manufacturing planes for the airlines.