Here is a tutorial by one of the best jugglers in the world. Now, here are my own disorganized ramblings.
(I am going to assume you are right-handed in the following) Before you try to learn three, you need to be able to do two balls correctly. Almost everybody will try to juggle two balls by throwing the first ball across from the right, and then passing the second ball directly from left to right to clear the left hand for the catch, causing the balls to follow one another around in a rough, counter-clockwise circle. This works fine for two, but to juggle three balls in this pattern, called a ‘shower’, is essentially impossible for a beginner.
The proper way to juggle two is to throw the first ball from the right, and then as it reaches its apex, to throw the second from the left to the exact same height. The balls should occupy the same plane, as it they were traveling along a wall in front of you, and the second throw should rise along the inside of the falling ball. In order to make this happen, each throw should be made from around the navel, and should be caught in line with the outside of the hip. The arms should be held parallel to the floor, and should _never_change from this angle by more than ten degrees or so. Do not reach up to catch a ball. Keep a relaxed stance, with good posture, but your upper arms should stay by your sides.
Learn to do this two-ball exchange starting with your left hand as well as with your right. If you find that the second throw is lower than the first, try throwing it considerably higher than the first throw for a bit, and then try to equalize them again. Make your throws to about eye level. Don’t try to wait for the first ball to reach the top before making the second, as you will then throw too late. Instead, make a rhythm in your head or aloud such that the two throws followed by the two catches make an even beat, 1, 2, 3, 4. Once you can do this both ways, try to keep it going at that same rhythm, right, left, pause, left, right.
Now, on to three. This can take anywhere from a few hours to a few days to learn. You will notice that after your right hand has made the first throw, it just sits there empty for a while. This is the ‘pause’ I mention above, and this is where the third ball goes. You do not speed up the pace at all, you simply keep throwing, right, left, right, left, starting with two balls in your right and one in your left. You will probably have trouble making yourself throw the fourth throw; whip it at the ceiling if you have to, but get the damn thing out of your hand. Don’t get stuck in the rut of stopping after three throws every time. It might also help to learn to start with your left hand as well as your right. Then, just try to get as many throws as you can; it definitely helps to count how many you can get, both to keep track of the rhythm and to have a personal best to beat. Take care, however, not to sacrifice form. Keep your hands by your waist. If you end up throwing forward and running after the balls, it is probably because you are letting your upper arms leave your sides, and if the balls are brushing each other near your hands, you are both throwing too late and are probably lifting your hands up. Remember to keep the balls in a single plane; most people try to lift the balls in front of each other.
This is how to juggle a ‘cascade’, the basic juggling pattern; it looks like an infinity sign. But there are thousands of other patterns that are infinitely more difficult and more rewarding. Juggling can be anything from a party trick to a hardcore sport, depending on where you want to take it. I would highly recommend that you look around on youtube (I can send you some links if you would like), and if you enjoy what you see, and are up for a challenge, you should try to learn some serious juggling, if not to impress your two year old then to impress yourself.