What is root movement?  Root movement is identifying what chord is moving to what other chord, regardless of inversion.  In other words, it is what Roman numeral is moving to what other Roman numeral. For example, when the tonic chord moves to the first inversion supertonic chord (I – ii6), the root movement is up a step (from I to ii).  Notice that root movement is not at all the same as the movement of the bass voice.  It only relates to the roots of the chords in the progression.

There are a few very common root movements that are consistently found in progressions in tonal harmony.  They always sound good, they move the harmony forward, and they also work well to provide resolution for sevenths of chords.  Learn these three, and they will help you know what chord to move to next, or what chord might make a good chord to approach any given harmony.  Just remember that the inversions of the chord play no part in the measurement of the root movement, only the Roman numerals.

In this description, I am choosing to use the “shortest” distance to describe the type of movement, since I have found that for most students this works most reliably. 

The most common movements are:

Up a 4th
Down a 3rd
Up a step 
I (i) goes anywhere

Each is discussed below in more detail:

Up a fourth: could also be described also as down a 5^th, but for most students it is easier (and shorter) to accurately count the distance upward rather than down.  Up a fourth is the strongest root movement – one obvious example is V to I.  Most of the time the interval a fourth up is perfect, but not always since there is one tritone in every scale. 

Examples of up a fourth root movement: V – I, ii – V, vi – ii, iii – vi, I – IV, III – VI, VII – III, IV – Vii, etc.  Remember that root movement just relates to the Roman numerals of the chords that are to be used; you may add inversions as you please to either or both of the chords.  This means that V6 - I and I - IV6 are also up a fourth root movements, even though the bass is not going up a fourth. It is only the Roman numerals that determine root movement.  When using this type of root movement, there is always one common tone, and the other two notes can move by step.

Down a third: This is the simplest and smoothest movement for voice leading: only one note is required to move in diatonic progressions, since there are two common tones.  Because of those two common tones, the second chord does not sound all that much different than the first.  Examples of down a third root movement: IV – ii, I – vi, vii^o – V, VI – iv.  Inversions can be used, so IV - ii6 and I6 - vi are also down a third root movments. Again, it is not the movement of the bass voice that we are interested in here, only the two Roman numerals in question.

Up a step: when describing this type of root movement, we mean the root moves up to the next letter name or step in the scale - it can be either a half step or a whole step.  This type of root movement is the most difficult for voice leading, since all three notes must move, and it is very easy to end up with parallels.  Use contrary motion to avoid this when both chords are in root position. Since the root goes up, the other voices should move downward.  Examples: V – vi, IV – V, I – ii, etc.  Again, inversions can be used as needed, which also may help avoid parallels.  For example, try using I – ii6 instead of I – ii, if parallels are a problem.

I (or i) goes anywhere.  In other words, the tonic chord can go easily to any other chord in the key, regardless of root movement.
 

You can use these principles also when resolving seventh chords:  these three good root movements (up a 4^th, down a 3^rd, up a step) will always provide you with the correct note needed to resolve a seventh of a chord. 

For example, if you are using a ii7 chord and want to know what chord would be good to put next, think about the step of the scale that is the seventh of the ii7 chord: in this case, the 7th of the ii7 chord is the tonic note.  Since sevenths of chords need to resolve down by step, the chord following the ii7 needs to contain the note of resolution one step down from the tonic note, which is the leading tone.  If you test the three root movements described above you will see that they each contain the required note:

The chord up a fourth from ii7 is V, and it contains the leading tone.
The chord down a third from ii7 is vii^o, and it also contains the leading tone.
Up a step from ii7 is iii, and it also contains the leading tone.

 
The "opposite" three root movements (down a 4th, up a 3rd, down a step) are not as common in tonal harmony, as they don't have as strong a feeling of forward "progression" towards the tonic.  Sometimes when these sort of progressions are found, the bass may 'mimic' one of the good root movments described above.  Once fairly common example of this is when you have a chord which moves down by step to the next chord (instead of up by step), such as the progression V - IV6.  Notice that although the root movement is downward by step, the bass voice moves up a step to give the progression a stronger feeling, similar to the up a step progression V - vi.