Airplanes generate only forward thrust (unless you're in a VTOL craft which we won't talk about). This forward force is what ends up generating lift due mainly to two particular, but separate, applications of physics.
First, wings take advantage of what is called Bernoulli's principle. Picture a wing: it is longer on top than it is on bottom due to the "chamber" of the wing. I.e., the asymmetric difference in the shape of the top and bottom of the wing. This causes the air that flows over the wing on top to be at a different speed than the air on bottom, since the air on top gets compressed. (Remember that in a river, if the river narrows, the current speeds up because you still have a stable amount of water to get downstream. If the volume status the same, higher pressure = higher speed, roughly.)
So, by Bernoulli's principle, the slower air on bottom has more potential energy than the air on top -- and that results in more outward pressure by the air, i.e., it pushes up on the wing harder than the air on top pushes down (which has more kinetic energy and less potential, due to the higher speed). Because the wing is not creating energy, the air is merely converting to and from kinetic/potential.
However, that's not the entire story of lift!
Wings are also angled, with their trailing edge down as compared to the leading edge. This angle is defined as the "angle of attack". Think about it like this: when you stick your hand out of the window of your car while driving down the freeway at high speed, you can tilt your hand up and "take off". The wind hitting your hand pushes it up because of the angle. This seems kind of straightforward, right?
If you continue tilting your hand closer and closer to the vertical, at some point it falls back down. You have "stalled" and are no longer generating enough lift to fly. Airplanes do the same thing, so they have to balance the lift they generate with the other factors.
Airplanes fly mainly through the combination of these two things: lift generated by the relative difference in airspeed above and below the wing, plus lift generated by the angle of attack.
I hope that is explanatory, and if I'm wrong on anything, someone else will come correct me!