While we generally think of airplanes when discussing flight, other flying machines, such as balloons and blimps, fly because of buoyant force, not because of lift caused by air pressure.
Archimedes' Principle
"Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by object."
Archimedes' Principle explains why some objects float in water and others sink. A stone sinks in water because the gravitational force acting on it is greater than the buoyant force; the stone does not displace enough water to support itself. For the same reason, a cork floats because it displaces a volume of water whose weight is greater than its own; its buoyant force is greater than its gravitational force. In technical terms, objects that "float" are called positively buoyant, and objects that "sink" are negatively buoyant. In general, an object floats if it is less dense than the fluid it is in, and sinks if its density is greater.
Since air is a fluid, Archimede's principle applies. However, since the only things less dense than air are other gases, virtually no objects "float" in air. However, by harnessing a gas and attaching it to an object, the gas's buoyant force is applied to the object, and the object rises. This is why balloons and blimps can "fly".
In this diagram, the balloon with volume 1m3 displaces 1.275kg of air, resulting in a 12.5 Newton upward buoyant force. Ignoring the weight of the balloon material itself, the helium gas inside weighs only 0.1664kg, resulting in a 1.63N downward gravitational force. The net force, therefore, is upward, and the balloon floats.