What makes an airplane fly?
Four Fundamental Forces

Weight

Weight is the force of gravity acting upon an object's mass.  The mass of an airplane is the sum of all parts that make up the airplane, including fuel and lubricating oils, crew/passengers, and cargo.  In a normal flight attitude, this force acts in the down direction, and it will always act towards the earth's center of gravity. 

The airplane's center of gravity, or CG, is the location where the aircraft's mass/weight is evenly distributed in all directions.  Optimally, the CG should be as close as possible to the airplane's center of lift.

Lift

In a normal flight attitude, lift is the force opposing weight, acting in the up direction.  It is produced by the interaction of the wings and fuselage with the surrounding air as the aircraft travels through.

There are three scientific principles at work in the production of lift as it applies to aircraft.


Bernoulli's Principle: As air travels faster, its pressure decreases.  A typical airplane wing is designed like a foil, where the upper surface is comparatively rounder than the lower surface.  Air flows faster and at lower pressure over the top of the wing than air flowing underneath it, actually arriving somewhat earlier at the trailing edge than the air traveling underneath the wing.  This pressure difference accounts for most of the lift, but not all.
The Coandă Effect: Airflow is attracted to nearby surfaces.  Air "hugs" the wing as the wing passes through it.  Additionally, since the airflow over the wing is faster than that below the wing, the net result after the wing has passed through is downward air movement.  As the wing angle relative to the initial airflow (the angle of attack) increases, the trailing airflow is increasingly deflected downward.  (However, too much angle of attack will result in disrupted airflow on the leeward side of the wing, causing the wing to stall, resulting in a loss of lift.)
Newton's Third Law of Motion: A force in one direction caused by an object is countered by an equal force upon that object in the opposite direction.  Air deflected downward off the trailing edge of the wing results in a simultaneous upward force on the wing itself.


Drag

Drag is the force of all friction acting upon the airplane, operating in the backward direction.

Drag comes in many varieties.  Skin friction is the resistance caused by moving a solid object (the plane itself) through the air.  Induced drag occurs at the wingtips, where the high pressure air below the wing flows into the low pressure region above the wing, creating inwardly rotating swirls of air (vortices) behind the airplane.  Additional drag is encountered as an airplane's speed approaches the speed of sound and beyond.

Thrust

Thrust is the force causing the aircraft to move forward through the air.  In most airplanes, thrust is produced by a propeller or jet engine.  This is Newton's Third Law in effect as well: as air is pushed backward, the aircraft moves forward.  This forward motion creates a flow of air over the wing, producing lift.  

When an airplane's thrust produced equals its drag, the airplane remains in level flight (barring any pilot control inputs).  When thrust exceeds the drag, the increased airflow over the wing creates additional lift.  When thrust is less than drag, the decreased airflow results in less lift being generated.