[Understanding this post probably requires a basic knowledge of seventh and related chords and extensions and alterations as used in "straight-ahead" (swing, bebop) jazz and mid-twentieth century American popular song harmony. The highlighted (and recommended) links will tell you what they are, and something of how they function in jazz harmony, though not the full story.]
A basic component of most jazz pianists' toolbox is the so-called "Bill Evans" or "rootless" or sometimes "left-hand" voicings. Each of the three terms is inaccurate. These were to some extent used before Evans came on the scene in the late 1950s/early 1960s, but he perhaps used them more extensively than others. (Wynton Kelly, Red Garland, and Ahmad Jamal are among those also cited as inclined to use them.) Along with McCoy Tyner, Evans also was a pioneer in using quartal voicings, which would probably be equally good candidates for associating with his name, but are not our topic here. A few of the "rootless voicings" contain the root of the chord, though most do not. And although they are commonly used in the left hand while the right plays melody, they may also appear in the right hand. "Four-note voicings" might be another term one sees used, though I'm not sure if that's as specific.
In some of the classic books on jazz piano playing (like Mark Levine's highly recommended "The Jazz Piano Book"), these may be introduced a bit too early, and some teachers may overemphasize them early on (especially to students who are already fairly aware of the basics of jazz harmony). Working pianists, who usually play in a rhythm section in which the bass takes care of stating the roots, like to use these voicings in order to stay out of the way of the bass line, and because they allow for more "color" tones, as found for example in standard extensions of 7th chords to include 9ths, 11ths, 13ths. (Some of these may be called "alterations", a term whose appropriate application I'm not completely clear about and am not going to get into here; it usually refers to a #11, b5 (enharmonically the same as a #11), b6=b13, b9 or #9, but precisely which notes are "altered" and which are just extensions depends on (what is considered to be) the harmonic context.) For learning jazz theory in a way that gets it into your ear (and fingers, if playing piano and not just hearing jazz harmony is your ultimate goal), I think it's best to practice four-note voicings with roots first. These can work for elementary solo piano playing, and for getting the sound of a tune including the roots fixed in your mind (play the "rooty" voicings in the LH, and the melody or an improvisation (yours, or a transcribed one) in the RH). Of course you can use these same possibilities with the Evans voicings, and you will find that many of them are the same as "rooty" four-note voicings for chords a third higher, so practicing the rooty ones first also helps with the Evans ones. I'll post on rooty voicings at some point, but here I'll discuss learning the more advanced Evans voicings, something I am in the middle of doing.
I'm using pianist Earl MacDonald's excellent post on how to learn these voicings. I recommend printing out both his post, and his pdf file with music notation and taking them to the piano. I won't go into detail, but just say a few things that might be a useful supplement. Two voicings, labeled "A" and "B", are given for each chord type. Frequently, though not always, the B voicing just involves taking the bottom two notes of the A voicing and making them the top two notes. For example, the first two chord types he considers (minor 9th, and [dominant] 13th) work that way. I tend to think of these kinds of four-note voicings as a pair of intervals (that between the bottom two notes, and that between the top two), separated by the interval between notes 2 and 3 (top of the bottom interval, and bottom of the top interval). Then I just think of the move to the other voicing as moving the bottom interval up an octave (or the top one down an octave, depending which way I'm moving it). It can help to keep in mind how the middle interval will change when you do that: e.g., for the minor 9th voicings, from a minor third for the A voicing (I don't think explicitly about this in this case, because the A voicing here is just a root position major seventh starting on the third of the chord we are voicing, e.g. Cm9 is voiced as EbMaj7) to a half step, or vice versa. The cool thing about these voicings is that when you want to move from, say, a B voicing to a voicing for the same type chord with the root down a fifth (very common root movement, with or without a change in chord type), you just keep the top two notes the same and move the bottom ones down a half step or a whole step. So again, thinking about the chord as a pair of intervals helps. Of course ultimately you want to get this into your fingers, and not "think" too explicitly. For example, to move the minor 9th B voicing to a minor 9th a fifth down, you go to the A voicing of the new chord, by keeping the top two notes the same as in the previous chord, and dropping each of the bottom two by a whole step. When you start incorporating the voicings into chord progressions, the chord type will often change, but since root movement down by fifths is common and important, you can frequently negotiate these progressions effectively by going from an A voicing for the first chord to a B voicing for the second, or vice versa, keeping track of which notes change and which stay the same. Often you will just move the bottom interval, or just move the top interval, which is nice. And if you've practiced root-included 7th-chord progressions, you might find some of the movements are similar, or the same, just used over a different root. I haven't done much along these lines yet, but obviously ii V7 I or the minor homologue, iiø V7 i, are the first ones to work on.
The basic construction principle for most of the voicings can be understood starting from the example of the minor 9th chord. The chord tones used are 3, 5, 7, and 9 (3 and 7 of course refer a minor third and minor seventh relative to the root, since this is a 9th chord; the 9th here is major). The A voicing is [3 5 7 9], B voicing is [7 9 3 5] (left to right going low to high in pitch). When a voicing has a natural 11th (enharmonically, 4th) it appears instead of the 3rd. (This happens with one chord type, the half-diminished chord with natural 11th.) When it has a 13th (= 6th) it usually appears instead of the 5th, in the above constructions. There is an exception to the 5 goes to 6 rule for the A form of the standard major (no 11th) voicing: the A form is a 6 9 voiced [3 5 6 9] (so one can think of the 6 as having been substituted for the 7th). A #11th, on the other hand (one chord type: the Maj7#11), is substituted for the 5 (the boppers used to think of the sharp 11 as a flatted fifth; thinking that way there is no substitution going on here; then again I don't think the boppers often added a sharp 11th to major chords). The Maj7#11 is also an exception to the rootless rule: it is voiced A: [1 3 11 7] and B: [11 7 1 3]. The other exception to the rootless rule is the B form of the standard (eleventh-less) major chord: it is a Maj7 with root, voiced [7 1 3 5], i.e. the major 7th and then the root-position triad, starting a half-step above the 7th. This pair of major voicings is the only one that doesn't obey the rule of putting the bottom interval on top while keeping the top interval as the bottom of the new voicing, to go from A to B voicing. Rather, the bottom goes on top, but the formerly top interval shrinks (if you must think this way) from 6 9 (a fourth) to 7 1 (a half-step) as it becomes the new bottom interval.
One could probably understand a bit more about the choice of particular types of voicings from the voice-leading properties they give rise to in common progressions (primarily major and minor ii V I or i type progressions). Curious is the omission of a voicing for the dominant 7th #11. This was a very important chord starting with bebop. If this reflects Evans' practice and not just MacDonald's predilections, I wonder if it's because Evans usually used a different type of voicing (quartal?) for this chord type?
If MacDonald's exercises seem time-consuming and difficult, let me just say that you can progress fairly quickly, and it's worth it. Here's a point from MacDonald that I really appreciate his emphasizing; it's crucial to remember, not just about this but about many, many exercises involved in learning to play jazz (and other musics, for that matter, e.g. scale practice):
Learning voicings is similar to learning to ride a bike. At first it is difficult, frustrating, and at times, painful. But once it is learned correctly, you never look back, and you can do it instinctually ever after.
A few comments on MacDonald's suggested learning routine. For all of the exercises, I've done them sometimes without sounding the root, but frequently with the root sounded in the bass. I think this is important to get the proper harmonic function of the voicing in your ear. Less crucially, I've done some of them with the right hand as well. Exercise number 8, taking the voicings down the circle of fifths with metronome (he refers to it as the circle of fourths; up a fourth is down a fifth, modulo octaves) is particularly crucial; I think this is where you'll really get the voicings memorized. Besides sometimes doing it with sounded roots, when I don't sound the root, I've been saying its letter name out loud. This also helps in better memorizing the circle of fifths, which anyone playing any music with essentially Western tempered harmony will want to do. Another point is that before working on each chord type, it is good to sound out the full chords, in root position, stack-of-(usually)-thirds configuration, and then compare this sound to the rootless voicing sounded with the root in the bass. You'll really start getting an idea of how extensions and alterations sound by doing this (especially if you sound out the lower seventh chord before adding extensions). You don't have to do this for every root (I haven't been), but it might be worthwhile too.
I have not yet made flash cards and done the "random roots" exercise. I've tried going up by fifths, as preliminary step toward getting away from the reliance on "muscle memory" and explicit thinking about the "lower the bottom two notes" trick for moving the root down a fifth while going from an A to B voicing, and I recommend it, as it's a cool sound as well. I'm ignoring his suggestion about completely mastering one chord type before going on to the next, in that I've worked quite a bit on the 13th chord without complete mastery of the minor 9th, but I think that's OK as long as you don't mix things up to much and really push on each type focusing primarily on one at a time.
Finally, the observation he asks you to try to ignore, that five of the chord types share the same voicing (just with a different root), is quite neat and important, an example of the general phenomenon that putting a different bass note under a given set of pitches in the middle or upper register can make an enormous change in the way they sound. Not only could it be used for reharmonization of a given melody, but I imagine it could be used (and probably is used) in composition, not just jazz but classical composition (many of these 7th, 9th, 13th, 11th, and 6th chords appear in classical music, especially Ravel, Debussy, Stravinsky, and Copland) to effect modulations, by changing the root under a given voicing and then treating it as if it has the new harmonic function, resolving it in some standard way. It would be neat to find---or create---examples of this.