An Aeolian mode formed part of the music theory of ancient Greece, based around the relative natural scale in A (that is, the same as playing all the 'white notes' of a piano from A to A). Greek theory called this simple scale the Hypodorian mode, and the Aeolian and Locrian modes must have formed different (perhaps chromatic) variations of this.
The term Aeolian mode fell into disuse in medieval Europe, as church music based itself around eight musical modes: the relative natural scales in D, E, F and G, each with their authentic and plagal counterparts.
In 1547 Heinrich Glarean published his Dodecachordon. His premise had as its central idea the existence of twelve diatonic modes rather than eight. It seems that popular folk music used the additional modes, but they did not form part of the official church repertoire. Glarean added Aeolian as the name of the new ninth mode: the relative natural mode in A with the perfect fifth as its dominant, reciting note or tenor. The tenth mode, the plagal version of the Aeolian mode, Glarean called Hypoaeolian ("under Aeolian"), based on the same relative scale, but with the minor third as its tenor, and having a melodic range from a perfect fourth below the tonic to a perfect fifth above it.
As polyphonic music replaced mediaeval monophonic church music, the folk modes added by Glarean became the basis of the minor/major division of classical European music, the Aeolian mode forming the natural minor mode. However, it would not be correct to refer to any piece in a now-traditional minor key as being in the Aeolian mode, which would imply that the style of the piece was modal, which is usually not the case with music in a minor key as understood today. In particular, the main (but probably not only) difference would be that the minor-key piece would frequently use the raised 7th degree as its leading note, particularly in the use of dominant-7th harmony, whereas the Aeolian piece would rarely or never use such a raised leading note, and its dominant chord would be a minor triad or minor triad plus minor 7th.
The Aeolian mode is the sixth mode of the major scale and has the "formula" 1, 2, 3, 4, 5, 6, 7. Its tonic chord is the submediant minor triad in the relative major key. For example, if the Aeolian mode is used in its all-white-note pitch based on A, this would be an A-minor triad, which would be the submediant in the relative major key of C major.
As the Aeolian mode forms the natural minor scale (also known as the descending melodic minor scale), it is among the most frequently used diatonic modes in western music. Tunes entirely in the Aeolian mode (i.e., those that do not also use the ascending melodic minor scale) are rare in classical music. However, they are common in many folk traditions, including Jewish and Israeli folk music, and the Israeli national anthem Hatikvah.
This chromatically-altered mode is also known as "Locrian sharp 2", Mode VI of the melodic minor scale, or the half diminished scale. It is frequently used in jazz and rock. The latter term is generally avoided by musicians, to avoid confusion with the diminished scales (see octatonic), and the half-diminished seventh chord.