A voltage is, definitionally, between two conductors. If you have one conductor, you have no voltages. No voltage, no power, nothing happens. Not terribly useful.
If you have two conductors, you have one pair (2C2), which allows for one voltage. We call this single-phase. Now we can actually make things happen, which is a substantial advantage over having only one conductor. But you can only make one thing happen; there is no possible variation in how the load can be connected. Put another way, there's only one dimension to the voltage: it's positive, or it's negative. One common problem is that if you hook up a single-phase motor directly to an AC line, you have no guarantee about which way it will spin, or if it will at all.
If you have three conductors, you have three pairs (3C2), which allows for three voltages. We call this three-phase. Now we can make three things happen, at different times. For example, you could have three electromagnets arranged in a circle and turn them all on in a sequence. Now we can guarantee that a motor will rotate, and in which direction. This is a substantial advantage over single-phase. Put another way, we now have two dimensions to the voltage; it's represented by a vector in a two-dimensional space. There are only two possible distinct arrangements of conductors ((3-1)!), which corresponds to the two possible directions of rotation.
If you extend this to four conductors, you have six pairs (4C2), so the next step is six-phase voltage. What advantages would six-phase have over three-phase? Well, now there are (4-1)! = 6 possible distinct arrangements of conductors, which means that if you're trying to cause something to rotate in a plane, you could hook things up in a manner that is inconsistent with that. So if you had a six-winding induction motor, it would be possible to hook it up in a manner that would vibrate horribly and rotate at half the normal speed, rather than just pick one direction or the other. That's not a plus.
But suppose that your rotor had three degrees of rotational freedom instead of one. With six-phase, and an appropriate mechanical arrangement of magnetic poles, you could induce rotation (roll, pitch, and yaw) in a floating spherical rotor of fixed position. Since such a thing does not exist to my knowledge, this does not really qualify as a useful application. (Maybe in a null gravity environment, where the magnetic poles are orbiting some body? But then, how are they all hooked up to the same six-phase AC line?) Of course, in a four-dimensional space, where we could have such a system and still translate all three directions of rotation to some other load outside our spherical stator/rotor arrangement, this arrangement could be hella useful.
Meanwhile, back in 3+1-space, I work in the world of industrial power electronics, and I've seen systems that use the kind of phase-shift transformers other answers have mentioned. As a matter of nomenclature, nobody I've talked to would describe using a phase-shift transformer to generate three more out-of-phase AC legs to be creating "six-phase". (By my math, you'd have fifteen-phase, but that's still not the language used.) When running three-phase through a rectifier into a cap, you get six pulses of current per cycle. For this kind of system, you'd get twelve pulses, so that kind of system would be called twelve-pulse.
(In general, the twelve-pulse rectifier is two six-pulse rectifiers. If you have two motor drives, you can connect their DC busses directly together and feed each with a different three-phase set. Or you can get a stand-alone rectifier for one set and feed its DC input into the remaining drive.)
If you're comparing a six-pulse rectifier to a twelve-pulse rectifier, with identical loads, each current pulse must be smaller to compensate for there being more of them driving the same load. This makes the overall current out of the line look somewhat more like a sine wave, meaning the harmonics are reduced. Ripple on the caps is also lower, but I've never known anyone to be terribly concerned about that.
Greater harmonics improvements can be had with an eighteen-pulse system and three rectifiers. (36-phase!) At higher voltages and powers, even higher numbers of paralleled rectifiers may exist. This document on a medium-voltage VFD line references a 54-pulse rectifier at 11 kV!
Three-phase power gives us one rotational degree of freedom, which is the limit of what is useful in a three-dimensional space.