It is Bernoulli's principle of lift that keeps airplanes in the air. Briefly stated, the principle is that the faster a fluid travels, the more the pressure that it exerts decreases. This is a principle with many applications outside of aviation, including the production of perfume atomizers and paint guns. Bernoulli's principle is not especially complex, but it does seem to run counter to common sense: things that move faster somehow seem as if they should be exerting more pressure, not less.
Consider water passing down a six inch pipe that decreases to a five inch pipe. The water must travel faster in the tighter pipe so that the same volume will get through in the same period of time. When the speed of the water increases, the energy that it had been expending by pushing on the sides of the pipe will be diverted to increasing its velocity. The water in motion only holds so much energy (stored in it when it was lifted) and this energy can either go to velocity or pressure.
It general then, the increase of velocity of a fluid, including fluid air, must decrease its pressure. For example, the rushing water from a shower head decreases air pressure in the shower stall and causes the shower curtain to be sucked in. Less benignly, the decrease in air pressure caused by a speeding heavy truck on the highway can cause an auto to experience a dangerous sway. Similar suction may draw a bicycle towards a speeding car. And commuter trains can, in the right circumstances, exert suction as they rush through a subway station. That's why the yellow lines are painted on platforms.
The wings of aircraft are shaped to enable Bernoulli's principle. They are relatively flat on the bottom, but arched on the top. This means that, in flight, air particles flowing up and over the wings will have farther to travel than the air sliding under the wings in a straight line. The air above the wing will have father to travel, so it will travel faster. (Yes, actually the plane is traveling, and the air is not, but everything is relative. From the vantage point of the wing, the air is moving.)
When the air above the wing increases its velocity relative to the air below the wing, it decreases the upper air pressure relative to the air pressure below the wing. This difference is what results in a pressure upwards: the lift. Lift is not, of course, a pressure from above, but an upward pressure from below, one that is unopposed by a pressure from above.
Obviously, Bernoulli's principle of lift only works if the aircraft in question is moving forward. If the plane were to stop, it would begin to fall. As long as it keeps moving though, Bernoulli's lift will keep it in the air.