With voicings means how a chord is arranged and the notes are spread out. Voicings are therefore deviations from the main variation of a chord and could for example be a version that is play on different octaves with one note omitted (removed from the chord).
Triad over root voicings
A simple triad voicing could look as this:
Instead for playing the regular C Major chord with C, E and G notes on one octave, the left hand plays the root note on a lower octave whereas the right hand plays the triad. This is called triad over root voicing and it add some extra bass and with that more depth.
Part over root voicings
When it comes to four-note chords, one method is to separate them in two parts. The left hand plays the root whereas the right hand plays the rest:
The Cmaj7 voicing above will add depth to the chord thanks to the wider span between bass and treble.
Closed and open voicings
Both examples above could be seen as open voicings since the notes are spread out over a wide range. The chord C11 could be play both with a closed and an open voicing.
The above picture shows a closed C11 voicing.
The above picture shows an open C11 voicing.
Closed and open voicings are often both present in progressions which can help creating smooth movement. Different voicings can be suited in different situations in which top note in chord match the melody note. Closed and open voicings can be also be suited differently depending on which top note in the chord that match a melody note.
A chord without the root may seem as a contradiction, but it is possible as long as other important notes in the chord are included. It can be seen as the voicing implies the chord, since the other important notes are still there.
So why play rootless voicings? One reason is that the root note can be replaced with another note which extend the chord and makes it more colorful. The chords can when be extended without increasing the number of notes, which reduce the fingers involved. In cases a bass player is involved, he will take care of the roots.