Jump to the part about Rameau.
We have come to think of chords as being defined as stacks of thirds. However, it's more accurate to say that chords are interpreted as stacks of third, or, better yet, chords are interpreted in terms of stacks of thirds.
Before thirds and sixths
Before there were "chords", as we understand them, there were "sounds happening at the same time": a melody accompanied by a drone, for example.1
By the ninth century, though likely long before, Europe was singing in parallel unisons, fourths, fifths, and octaves. When singing in parallel fourths or fifths, in order to avoid tritones, the accompanying voice might change briefly to a drone before continuing.2 This technique would produce intervals other than the "perfect" ones, but the focus was on maintaining perfect intervals.
The earliest known European polyphony also revolved around unisons, fourths, fifths, and octaves. The occurrence of other intervals was incidental to the movement of the voices, and the aim remained to arrive at perfect consonances.3
The coming of thirds and sixths
Initially, thirds and sixths were considered dissonant. Although allowed in early polyphonic music, their presence, often in parallel, was a somewhat distinctly English preference.4 By the 1300s, thirds and sixths were increasingly used across Europe, as dissonances requiring resolution to a perfect consonance.5
The arrival of thirds and sixths
The European Renaissance brought major aesthetic change to music. Among the most important, the acceptance of thirds and sixths as consonant,6 and a new focus on the polyphonic voices as being of equal standing (as opposed to earlier polyphony, in the which the voices were generally subservient to the primary melodic voice).7
Thirds and sixths
With the idea of "counterpoint" now developing, "chords" were still understood in terms of the melodic progression of voices. @Athanasius gives an outstanding description of this in answering the question Did continuo players consider figured bass as “interval symbols” or “chord symbols”?
Finally, we get to thirds. In 1722, Jean-Phillipe Rameau published his Treatise on Harmony (Traité de l'harmonie). Based on this treatise, Rameau is credited with the idea of chords containing a root, of triads and seventh chords being the basic units of harmony, of inversions of chords as being "the same chord", and of chords being stacks of thirds.8
1"Europeans probably performed music in multiple parts long before it was described. The simplest type, singing or playing a melody against a drone, is found in most European folk traditions and many Asian cultures, suggesting it dates from antiquity." Burkholder, J. Peter, Donald Jay Grout, and Claude Victor Palisca. 2006. A history of Western music. New York: W.W. Norton. Page 88.
4Ibid., 111, 114.
6Ibid., 157. "The core of the fifteenth-century style was a new counterpoint, based on a preference for consonance, including thirds and sixths...."
7Ibid., 159. "A striking change occurred during the second half of the fifteenth century, when composers moved away from counterpoint structured around the [primary melody] and toward greater equality between voices."