What makes a chord "stable"? What makes a chord "unstable"? Just saw this in my piano book, but it doesn't really explain what it means.
The concept of tonality is partially based on the idea that certain chords "want" to go to other chords. For instance, the dominant (V) wants to go to the tonic (I), mostly because it has the leading tone (scale degree 7). A more complex example is the augmented sixth chord. The augmented sixth is VERY unstable, because it has two notes (the flat-sixth and the sharp-fourth) that both want to move to the dominant (i.e. scale degree 5). You can hear this instability in the chord. So stability is primarily based on the chord's purpose in relation to the tonal structure you're in, but at the same time something like an aug-sixth is never going to feel stable.
As mentioned in a comment by Caleb Hines, there is no clear-cut definition of the concept of stability in music. When talking about chords, the two notions of stability that I consider most important are the stability with respect to a given key, and the stability of a chord without any context.
In a given key, the tonic is perceived as the most stable chord, and the dominant is perceived as unstable because it needs to resolve to the tonic. The supertonic may be perceived to depart from the tonic, moving to the dominant, etc. However, outside of textbooks, this has to be taken with a grain of salt. If you analyze actual pieces you will find out that this concept of stability is very relative, and that chords don't always move in the most expected way.
Another concept of stability takes the chord out of its context, and only looks at its degree of consonance or dissonance. In jazz theory, any chord with a perfect fifth (i.e. also a dominant seventh chord) is considered a stable chord. Chords with a diminished fifth (e.g., a diminished or half-diminished seventh chord) are considered unstable. The practical consequence of this notion of stability is the fact that only stable chords can be preceded by their respective secondary dominant chord, because they convey some feeling of resolution.