These kites have four lines attached to two handles. Although you can attach 4 lines to anything and get it to fly, I think the trick is to have it flying in a fast, controllable and predictable way. All of the quad line kites I have seen or flown were designed from the outset to do that.
In terms of flight a true quad line kite will fly forward and backwards, with possibly the ability to slide it from side to side. Given practice and a favourable wind you can achieve a very fine level of control. It is even possible on some of the kites to do some of the more radical tricks usually found on the dual line kites (such as Axels)
Quad line kites are rarer than dual line as you only tend to find them in proper kite shops, not normal seaside fun shops for instance (at least the shops I've seen) and they cost more than starter dual line kites (the cheapest new quad line kite I'm aware of is the HQ Magic Quad at around £60).
A lot of people believe that these kites are much harder to fly than dual line kites - that is true, to a point. With a dual line kite it is intuitive on a basic level on how to fly - pull left and the kite goes left, right and the kite goes right. A quad line kite you have to push the top of the left handle forward to turn left, push the top of the right handle to turn right and so on. But since you can fly forward or backwards it's a lot harder to equate your hand movements to what is happening to the kite. And most quad line kites don't like the normal dual line hand movements (it's been said that people who fly a dual line kite have a harder time learning than people who have never flown).
It just takes a bit of time and practice - like everything it's not hard once you can do it!
These kites look like a 'W' shape - the flat top is the 'leading edge', that is the main direction of travel. They are capable of (obviously!) going forward, but can move backwards and side to side. Added to that they can change speed and direction very quickly and can spin tightly on the spot. The Revolution kites come in a number of different types - the 1.5SLE is the 'standard', the cheaper EXP looks the same, just isn't as fast of accurate in flight, the more expensive Supersonic, Shockwave or Blast are flatter and more rigid (they are like a 'W' with the bottom points cut off) and are much faster and powerful.
Of all the Quad line kites these are now perhaps the most common shape.
These are quad line kites that aren't made by Revolution Kites, but act in nearly the same way. So to avoid the copyright issues they adopt slightly different shapes - usually with pointy bits on the top and bottom - looking much more symmetrical than the standard Revolution design. Common example are the HQ Magic and Hypnotic Quad kites.
I have heard they don't fly as well as Revolutions as they lack the Revolutions leading edge, but as yet I haven't had a chance to fly these so I can't comment.
Obviously you can attach any number of lines to a kite - some designs are very strange - such as the Synergy Deca series of quad line kites. These are 3D structures - part kite, part engineering marvel, part work of art and are flown in the quad line style - they can't do everything a Revolution can, but do have a few extra tricks they can do.
According to some Internet sites you CAN rig up a standard Delta to be four line kite. Whilst this would work, it is doubtful weather they are as manoeuvrable as a dedicate quad line design, such as the Revolution.