The prevailing theory is that the "function" of the seventh is to provide a dissonance with the basic triad and give it its character with regards to other chords in the chord progression. The idea therefore is that the seventh says whether the chord is "moving" or "staying", which is a little different than "stable" and "unstable" because major sevenths can sound very unstable and dissonant in their own way.
The reason for this somewhat lies within the classical usage of sevenths, and somewhat in modern usage. Traditionally, the minor seventh above the root of the chord, like in minor 7 or dominant seventh chords, has a strong desire to resolve down a half step to a tone in in the next triad, usually a V dominant seventh chord (if its a II-7) or a I chord (if its a V7). Because the minor seventh is a dissonance with the root, traditionally, it must resolve. In contemporary music, it doesn't need to resolve - the dissonance is accepted as "color" rather than being unstable - HOWEVER, there are still resolution patterns that are left over from classical theory, such as the desire for it to resolve down a half step. This is why people sometimes classify minor sevenths as "moving", they don't tend to sound very final. Jazz pianists traditionally won't voice a tonic minor chord with a minor seventh (no I-7) because of this - it doesn't sound like its resting enough for it to be tonic. There might be the tacit implication that the chord will then resolve down a fifth to IV7. Generally they'll play a I-6 or a I-(maj7) instead, which, although probably more dissonant, sound more like "home plate".
The theory of major sevenths follows similarly, except in the opposite direction. Because the major 7th is the leading tone in the key, in classical theory there is a strong need for that note to resolve up a half step to the root of the chord, instead of down a half step to the third of the next chord in the cycle of fifths. In this way, it is "staying", because the seventh's desired resolution is up a half step to the root of the same chord other than a note in another chord. It still can be plenty unstable as a dissonance (min(maj7) chords, maj7(#5) chords), but it means the chord doesn't necessarily have that subtle pull somewhere else.
Now, sixth chords used to be pretty common in popular styles because unlike the seventh, the sixth doesn't create a dissonance with the tonic. In traditional thinking, the dissonances are 2nds, 4ths (remember, traditionally, 4ths sound fine to our ears today), and 7ths, and everything else is consonant (basically). When you add the major 6th, you have a consonance with the root, making it both "stable" AND "staying". Frequently, in the 30's and 40's, the tonic minor chord was voiced as a minor 6 (minor triad plus major sixth), and the major chord was almost certainly a major 6 chord. As time progressed, however, people desired the color of the major 7th and minor 7th (and the subsequent extensions beyond the octave) more and more, and so 6th chords aren't used as often. In Brazilian and Afro-Cuban music, however, even today tonic major 7ths aren't used often.
