Airbus vs Boeing
Boeing and Airbus are the two largest manufacturers of commercial aircrafts in the world. While Boeing is american and has headquarters in Chicago, Airbus is a subsidiary of EADS (European Aeronautic Defence and Space company).
Boeing was founded in Seattle, Washington in 1916 by William E. Boeing. The name of the company was originally Pacific Aero Products Company. On May 9, 1917, the company became the Boeing Airplane Company, after becoming independent. Airbus began in 1970 as a consortium of European aviation firms to compete with American companies such as Boeing, McDonnell Douglas, and Lockheed.
Unlike Airbus, Boeing produces military aircrafts and spatial material besides commercial airliners. Boeing owns also the company McDonnell Douglas. Airbus produces only commercial aircrafts ( nope A400M is not commercial). But they are the most sold in their field. Indeed, Airbus produces around half of the world's commercial airliners. Currently the company produces the following aircrafts : A318, A319, A320, A321, A330, A340, A350, A380 (which is the biggest commercial aircraft in the world). And Boeing : B737, B757, B767, B777, B747, B787 Dreamliner plus military and spatial products, weapons, rocket launchers...
 747 vs A380
Boeing made the 747, while Airbus made the A380. Both planes are jumbo jets, but the A380 is larger and more fuel efficent than the 747. Boeing is building a new 747-8 to compete with the A380, and the 747-8 is a little larger than the 747-400. Also, the top floor of the 747-8 is the size of a 737 (Megafactories), and the 747-8 is more fuel efficient than the 747-400, and it uses the same GEnx engines the 787's use. (The GEnx engines have some wave pattern at the end of that engine, so they are quite distinct)
 787 vs A350
The Airbus A350 XWB is going to compete with the Boeing 787 Dreamliner. There are 6 787 being built, but there are no A350s being built yet.
 737 vs A320
The Boeing 737 and Airbus A320 are short to medium range airliners. Budget airlines always buy them.
 767/777 vs A330/A340
The Boeing 777 200LR broke the world record for flying the world's longest routes, so it is good. The a330 is a smaller jet than the 777, and some budget airlines like AirAsiaX bought it. Note that the 767 isn't fuel efficient when compared to the 787, and the a340 has four engines!
 A little note...
Airbus planes use a small joystick at the side of the plane, opposed to the (traditional?) central yoke. Also, according to a book, (QF32 I think) you can pull back on the joystick, then return to neutral, and the plane will keep climbing at the same rate.
Also, the biggest cargo plane airbus has is the Beluga (I think). Boeing has also got an oddity like that, the dream lifter (carries Dreamliner parts)
Mega factories says that Boeing's factory can fit Disneyland inside (with parking space) The additions I added (the GEnx engines, for example) were not from the original author. Please change any mistakes. ;) (Sorry for the unofficial nature of the changes I added.)