For an understanding of the various ways of explaining lift, see lift. This force can be harnessed to lift an aircraft, or, in an inverted position, to hold a car or other vehicle to the ground. Airfoils are also found in propellors, fanss, and turbines.
It is important to note that any thin object at an angle of attack with respect to the airflow, such as a flat plate or a bridge, will generate lift, there is nothing "magic" about the shape of an airfoil. However, the airfoil shape ensures that lift is generated with the minimum of drag, so it is important for efficiency.
The optimal design of airfoils has been much-studied, and is a key element in aerodynamics. Different applications will call for a different airfoil - there is no one "true" airfoil design. Various systems have been devised to describe and characterise airfoils - the most common and prevalent is the NACA system. Before this, various ad-hoc systems were used. An example of a general purpose airfoil that finds wide application, and predates the NACA system is the Clark-Y.