Planes and helicopters fly due to their wings. If a plane loses its wings, it falls, period. But how do wings actually enable planes to fly? I will elaborate on that here.
As a wing or rotor moves through the air, it splits the air that flows into it into two parts: one stream flowing above and one below. The upper stream will naturally travel in a straight line, but is pulled down by the curve. This causes it to occupy more space, lowering its pressure. Conversely, the lower stream is squashed by the angled bottom of the wing. This causes it to occupy less space and thus produce more pressure. This combination of low pressure above and high pressure below pushes the wing upwards, along with anything attached to it. In fluid dynamics, this phenomenon is known as Bernoulli's principle. This phenomenon can be experienced quite easily on the ground. Anytime your travelling fast in a car, open the window and stick your hand out at a slight angle. You will be able to fell the air pushing your hand up.
The angle of attack of a wing also plays a significant role in increasing lift. The steeper a wing is, the more lift (and drag) it generates, until a point. The steepness of a wing lifts a plane because it pushes both streams of air downward into the ground, producing a stream of air called a downwash and pushing the wing and the plane into the air. This is also why planes with non-aerofoil-shaped wings and upside-down wings can also fly. Helicopters fly because their rotors produce a vertical downwash which pushes the whole system upwards. Planes also produce a downwash, though it is angled and thus a lot less noticeable.
A wing which is angled correctly and moving fast enough will cause both streams of air to split and then meet each other at the end. A wing angled too steeply will prevent the two streams from meeting, reducing lift and increasing drag, leading to what is known as a stall. This is why planes cannot take-off too steeply i.e. with too great an angle of attack. (Right): A diagram showing the effect of various angles of attack on lift.