How can such a heavy machine as a plane hang up in the air? Or how come paper planes fly? Aren't they subject to gravity?

The forces in question can counterbalance each other. The purpose of a jet engine on a plane is to create a force that will counterbalance gravity. The force created is a little bit smaller during the touch down or descent, and a little greater when taking off or gathering height. The jet creates aerodynamic lift which is responsible for ascending. When the plane gathers speed on the runway, it cuts the air with the edges of its wings. The shape of their edges and the inclination angle of the wings result in a difference in pressures between the top and bottom surfaces of the wings. The flow round path is longer on the top surface of the wings what results in lower pressure above and aerodynamic lift facing upwards.

Pressure differences often appear between the inside of buildings and the air that surrounds them. The flow of air to balance this difference creates draughts. The jet plane creates such a drought - a driving force.

According to Bernoulli's Law a faster flow causes a lower pressure exerted by the flowing fluid. Therefore, planes have a convex profile of their wings - the convex surface is the upper one. In this respect they resemble flat-convex lenses in their shape. The curved upper surface provides longer flow of air as compared to the lower part of it. The difference between a smaller pressure on the upper surface and greater on the lower one gives a resultant net force called the lift of a wing, which should balance the weight of a plane.

Usually the centre of gravity of a plane is not located right under the (bigger) wings. Horizontal fin at the rear (which allows for steering a plane) provides some supplementary upward thrust. It is the resultant lift that balances the weight of the plane.

Even a plane-shaped wing positioned at an angle to the horizon, as a plane during a take off, provides lift force by air resistance. But such force is considerably weaker compared to the Bernoulli's effect lift.