Hollow pipe-like objects, such as a hollow reed or bamboo, must be the oldest air instruments. Humans most likely blew into hollow objects to while away time, which is perhaps one way music began.
At some stage in the history of humankind the narrow side of an animal horn (which is hollow by nature) was cut off, and blown in the same fashion as hollow reed pipes.
Some time in the distant past it was discovered that different values of diameter and length of pipes result in different pitches. Later (here determined logically on the basis of complexity, and not by archaeological finds) it was discovered that if holes are made into a pipe, by closing and opening the holes different pitches could be made using a single pipe, and the flute was born. For many thousands of years fingers were used to cover the holes, until technology developed to make possible the closing and opening of holes mechanically. This made possible using stops that could cover holes larger than fingers, and consequently produce soundwaves of lower frequencies than was possible before.
The diameter of reed and bamboo instruments is more or less consistent along its length. The diameter of an animal horn typically flares out from one end to the other. To distinguish pipe instruments that are artificially made to flare out from instruments that are straight (or conical) pipes, the name shawm is used. The oldest known shawms date back to the times of the Ancient Egyptians. A shawm is basically flared pipe, and it may be regarded as an artifical horn.
All modern air instruments are later variations of these basic principals. The droning horn may be conceived as just a very large hollow bamboo or cane reed stem. The bugle is a bent pipe, which is easier for handling. The flute is a pipe with holes. "Reed" instruments (such as the saxophone, which should perhaps be more neutrally called "instruments with resonating modifiers") are pipes with resonators added to the air inlet side of the instrument. Originally the reeds of such instruments were indeed a piece of reed, but more recently other materials are used as resonators. Some instruments (such as the oboe) have more than one reed.
Instead of using holes in a single pipe to be used to change pitch, pipes of different lengths and diameters can be strung together into a single instrument. The panflute is perhaps the most ancient such instrument, while more modern variations of this multipipe instrument would be the pipe organ, but also the harmonica and accordion. Panflutes are found in many cultures and with many different names. Reeds could be added to these instruments. In essence, a harmonica and an accordion is a multi-pipe instrument with reeds.
Air from human lungs was originally the source of energy to initiate the soundwave. When mechanical methods became possible, they were also used. Bellows are the most common contemporary alternative source to move air. Bellows are used in accordions and concertinas, as well as in pipe organs.