Voice leading, and why the smallest movement is usually the best
How each voice moves from one chord to the next; common tones stay put, the rest move by step wherever possible, and the dominant's tritone resolves by contrary motion.
A chord is a set of notes, and a progression is a sequence of sets. But the ear does not hear sets: it hears voices, lines going from one note to the next. Voice leading is the study of how each line moves from one chord to another, and its central rule fits in a sentence: move each voice as little as possible.
Blocks and lines
Play C major and then F major, both in root position, with the same three notes rising as a block. All three voices leap a fourth together. Now play F major keeping the C where it was and moving only the other two: E rises to F, G rises to A.
The chords are the same; the second passage sounds like music and the first like a staircase. The difference is that in the second one voice stayed put and the others moved by a step. That is the principle, and everything else follows from it.
The three practical rules
Common tones stay put. If two chords share a note, the voice that has it does not move. C major and A minor share C and E; only the G moves, to A.
The rest move by step. A voice that has to change goes to the nearest note of the next chord, by tone or by semitone, and leaps only when there is no alternative.
Voices neither cross nor move in parallel fifths and octaves. The upper voice stays on top; and two voices a fifth or an octave apart do not move in the same direction keeping that interval, because the ear fuses them into one and the texture loses a voice. It is the most argued-about rule in classical harmony, and its reason is this: preserving the independence of the lines.
The tritone resolves by contrary motion
The most important case is the dominant. G seventh has B and F, the tritone, and the two notes resolve in opposite directions: B rises a semitone to C, F falls a semitone to E. Two voices, two semitones, contrary motion, and the tonic arrives with its root and third already in place.
Doubling notes is the consequence: a triad has three notes and a four-voice chord needs four, so the root is doubled. In four voices, the complete resolution of G seventh to C major leaves the arrival chord without a fifth, or makes a voice leap; classical writing accepts both and generally prefers the tonic with no fifth and a doubled root.
Inversions come from here
Voice leading is the reason inversions exist. If the bass is a voice and ought to move by step, it cannot leap from root to root; so the chords invert so that the bass can walk.
The bass's descent, from C to D, is a line, and the slash chords are the ones the line requires. Seen this way, inversion is not a choice about the chord: it is a consequence of leading the bass.
In popular music
Whoever plays keyboard or guitar leads voices without writing anything down: it is the difference between "jumping from chord to chord" and "changing chord with the hand still". A well-played II–V–I moves each voice by a semitone, and that is why it sounds smooth; the same II–V–I in three root positions leaps and sounds like three blocks. The chart does not write this, and the hand decides it.
D minor seventh, G seventh, C major seventh, the upper voices moving by semitone or staying put. It is the leading the ear expects, and it is why the sequence sounds inevitable.
What to take from here
Voice leading is the movement of each line between two chords, and the rule is the smallest movement: common tones held, the rest by step, no crossings and no parallel fifths or octaves. The dominant's tritone resolves by contrary motion, and inversions exist so that the bass can walk. At the instrument, it is the difference between leaping from chord to chord and changing chord with a quiet hand.