Daniel is an Army Aviation officer currently serving as a UH60 Blackhawk Pilot in Texas. He has a bachelors degree in history.
Back to Newton
Firstly, flight can simply be explained by one of history's most famous scientist: Sir Isaac Newton. Newton's three laws of motion are responsible for explaining many things in our world from simple movement of objects to complex things such as flight. His first law of Inertia simply states that an object at rest will remain at rest and an object in motion will remain in motion until something else acts upon it. Something needs an external force to start or stop moving (for example if a ball is rolling it will eventually slow and stop due to friction, where as it would take longer for an ice cube to stop sliding because there is less friction).
The Second law about Acceleration says that an object in motion will require a force directly proportional to its mass and force to produce a change. For example, if you try to push a large truck and apply the same amount of force you would use to push a small car, the truck wouldn't move as much as the small car would due to the difference in their mass.
Newton's Third law of Action and Reaction may be the most important in not just flight, but most things in physics. Basically, every action has an opposite but equal reaction. For example, if you are ice skating and push yourself off a wall you will move across the ice with an equal amount of force you applied to the wall. On a helicopter the rotor blades push air downwards causing the air to push back upwards on the blades and aircraft.
Bernoulli's Principle and the Venturi Effect
In order to understand Bernoulli's Principle of Differential Pressure, first we need to understand the Law of Conservation. The Law of Conservation states that energy cannot be created or destroyed, therefore if energy enters a system the same amount of energy must exit that system. Bernoulli's principle describes the relationship between fluid pressure and velocity which goes hand in hand with the Venturi effect which explains that a decrease in internal pressure increases velocity flow. The best example for this is if water is flowing through a garden hose the flow rate remains constant until a section of the hose is compressed, which would increase the flow rate (velocity) of the water going through that section (which is why water sprays farther if you cover the end of a hose with your thumb).
This explains how an airfoil (a wing or rotor blade) creates lift. As air hits the front of the airfoil it is separated and most is turned over the top of the airfoil. Newton's Third Law explains how the turn of direction of the air causes the air to "push" in the opposite direction and create lift. As the velocity of the airflow increases static pressure decreases, and since air usually travels farther over the top of the airfoil due to its design, there is a greater decrease in static pressure on the top than the bottom and thanks to Newton's Third lift is created.
Now that we have lift, where does the object go? To determine this we use vectors, which are simply depicted quantities of direction, size, and magnitude. The four main vectors of an aircraft in flight are Lift, Weight, Thrust, and Drag. Lift, as we talked about, is simply that: the force that is lifting the aircraft into the air. Weight is also self explanatory : its the result of gravity acting upon how heavy an object is. Thrust is the force propelling the aircraft in a certain direction, and drag is the resistance to the object moving.
As one vector changes, the opposite must also change. For example if thrust is increased causing the aircraft to go faster, then drag also increases due to the resistance from the aircraft pushing through the air. These are controlled by airfoils such as wings on an airplane or rotor blades on a helicopter. Pitch refers to the up and down axis, and aside aircraft attitude it is used to describe the up or down angle of an airfoil. If an aircraft pitches up, the airflow creates a greater pressure differential and produces more lift, and the increased lift becomes greater than the force of the aircraft's weight causing the aircraft to rise. Conversely, if an aircraft accelerates (increases thrust) the resistance of the air pushing against the aircraft (drag) also increases due to Newton's Third Law. When an aircraft is flying straight and level in a constant velocity, all forces are equal to each other. When the aircraft changes its flight profile (accelerate, descent, etc.) one force vector is acting greater than its opposite.
The best example is when you stick your hand out of the window of a moving car. The pressure you feel of the air pushing your hand back is drag, and when you push against it you are creating thrust. If you lay your hand flat there is little drag pushing since there is a smaller surface are, and if you turn (pitch) your hand slightly up you will see it rise from the differential pressures on the top and bottom of your hand creating lift, and vice versa if you turn your hand downwards.
That's How Things Fly!
This simple physics review shows how basic principles that we see in our every day lives causes the seeming complex phenomenon of flight. We can see these principles from watering our yard (Bernoulli and Venturi) to sticking your hand out of a car window (Forces and Vectors). While flight can get a lot more complicated when it comes to maneuvering an aircraft certain ways, I hope this helped explain the basics and show the simple reasons of why things take flight.