Essentially, there are four basic forces that act on a plane in flight. These are lift, weight, drag, and thrust:
The lift on an airplane is caused by differences in air pressure above and below the wing; this is called Bernoulli's principle, and is discussed in further detail on this site.
In short, the airfoil shape of an airplane's wing causes the air pressure under the wing to be greater than that on top of the wing, causing an upward lifting force on the wing; if the lift is greater than the weight of the plane, the airplane flies.
Those flying machines that don't utilize Bernoulli's principle for lift use buoyant force. Archimede's principle states that there is an upward force on an object in a fluid equal to the weight of the fluid it displaces. So objects that displace a volume of air with greater weight than its own (i.e. objects less dense than air) can fly.
Gravity, one of the four fundamental forces, is the universal attraction between masses. The force of gravity is given by the equation F=Gm1m2/d2, where m1 and m2 are the masses of the two bodies, G is the gravitational constant G=6.67x10-11 , and d is the distance between the two bodies. Here on earth, the force of gravity acting on an object is given by F=mg, where m is the object's mass, and g is the acceleration due to gravity (about 9.8m/s2).
Weight is a measure of the gravitational force on an object, and flying machines are, of course, subject to gravity. Weight holds objects to the earth, and counteracts lift on an airplane and the buoyant force on lighter-than-air aircraft.
Thrust is the force propelling an object. Thrust can be generated in a number of ways--propellers and jet engines are by far the most common. Thrust not only must overcome drag, but in the case of airplanes must give the airplane the necessary velocity to generate enough lift to fly.
Drag is the force resisting the thrust. It is a friction-like force between an object and the fluid it is moving through. Drag is dependent primarily on the surface area of the object exposed to the fluid; therefore, a more aerodynamically shaped airplane will encounter less resistance due to drag.