The Weave is a great three ball juggling move. Its sheer brilliance is down to several factors. For a start, it takes one of the nicest moves, Columns with a Carried Ball - which looks easy, and really really isn't, and adds to it something which is relatively easy and looks impossible. The effect of the combination of the two is a boggled-audience: something which jugglers like, on the whole. The move is, essentially, a one-handed one: the second piece of brilliance, then, is that you do it with one hand half the time, and the other hand for the other half; you swap between the two.
Okay. Juggle a column with your right hand: use two balls, one in the middle of your body, and the other on the right side. As the right ball goes up, the middle one comes down, is caught and thrown. The right hand goes to the right, and catches the right ball coming down; it throws it straight back up. Nice and simple. The left hand, however is doing this: as the middle ball comes down, the left hand goes over it, and under the right ball. It then goes around the right ball as it is coming down (leading with the back of the hand - hard to describe: do it - it'll be obvious), and heads back towards the left of the pattern, under the middle ball, which the right hand has just thrown up.
You could just keep going - throwing the columns and weaving the left hand in and out. It can be far flashier, though. Practice this until it's comfortable. Now practice it with the left hand throwing the columns, and the right hand doing the weaving.
Tricky stuff now. You need all three balls. Throw a column with the right hand and weave round it with the left, carrying the third ball. When the left hand gets back to the left of the pattern, grab hold of the far right ball with the right hand, launch the left ball on the left side, straight up (in a column) and carry the right ball across over the middle ball to loop the far left ball. Throw the middle ball back up in its usual column routine, but this time with the left hand (the right's still carrying). Bring the right hand over the far left ball as it's coming down and bring it under the middle ball on its way up. Catch the far left ball, and, when the right hand gets back to the right side of the pattern, launch the ball it's carrying up to start the right column off again.
It is insanely difficult, and it requires a great deal of practice. Arms are crossing each other, carrying balls and doing weaves. It also makes great demands on both hands - I have great difficulty making the left columns accurate for the right hand to weave through (as I'm right handed). But it's great fun - and incredibly impressive to watch when it's done smoothly. I don't make any claims to be able to do it nicely, and I've been practising it for years. But, as I've said elsewhere, I'm a really slow juggler.