How do you explain spinning tops to a nine year old? Why don't spinning tops fall over? (The young scientist version)
My nine year old son asked me this very question when playing with his "Battle Strikers" set. Having studied Physics myself, I am very keen to encourage him to take an interest in Science and I am delighted when he asks me such questions. In this case, however, I'm stumped. How do I explain why a spinning top doesn't fall down without going into the mathematics of angular momentum? 
I thought the following post would help, but the answers would just bamboozle him I'm afraid. 
Why don't spinning tops fall over?
So, how do you explain spinning tops to a nine year old? 
 A: Momentum as an english term is somewhat abstract, imho. Personally I consider the Russian equivalent  "момент количества движения" (roughly translated as the moment of the amount of motion) to be much more illuminating to a layman. Disregarding the term moment you can describe angular momentum (and linear momentum) as an amount of motion, since quantities of things is something a 9 year old should be fairly familiar with, I propose pivoting explanations around this interpretation.
I am of the opinion that self-discovery is always better than lecture or explanation, so I try to facilitate the child coming to the conclusion (through leading questions and hints) rather than just telling them the answer
Step 1
Give the child a top and ask them to come up with some way to give a number to describe the spinning. Presumably they will come up with something akin to angular velocity, the number of rotations in some time, etc. 
2 Proceed to ask the child to assign a direction to the spinning. Presumably their intuition will jump to clockwise and counterclockwise. Lead them into the right direction by challenging them to more precisely state the direction of the spinning. The answer you are looking for is "clockwise around an axis of rotation that is pointing up". Leading questions like "what is the top spinning about?" "Which way is up for the top" can lead to this kind of understanding
3
Tell the child that the direction and the number they came up with in the previous parts (disregarding mass for the moment, since it is fairly just a scaling factor) can be thought of as an "amount" of spinning, the same way that saying "there are 5 apples" is an amount of objects. The analogy being that you have a number and a qualifier, apples corresponds to a direction, 5 corresponds to the angular velocity.
4
Ask the child if 5 apples can become 5 oranges magically. Use this line of reasoning (and the analogy) to explain that 5 clockwise around that axis pointing up, can't magically become 5 clockwise around the axis pointing sideways. 
5
Asking the child if they can identify if anything is actually touching the spinning top. Conclude by asking them the leading question "If nothing is touching or bothering the top, why would the amount of spinning change?"
After a theoretical discussion it is good to have a physical demonstration - Lubos's suggestion of using a bycicle wheel is perhaps the easiest way to demonstrate the fact that "things want to keep spinning the same direction* in a way that the child can feel. 
A: Your 9-year old son?  OK, I'll give it a shot.
Warning: I'm gonna play fast and loose with my terminology, and use the word "force" when  "impulse" or "torque" or something else might be more appropriate.  I'm trying to figure out how to relate things to a 9-year old, and I think things can be conveyed more simply by describing everything in terms of force (which is something he can feel).
First, you want to get him familiar with the idea of linear momentum.  If you have a dolly with rotatable wheels, sit on it and have him push you around.
Point out that it takes force to get it moving and to stop it moving.  Then, have him push you around in a circle (or an arc), so he can understand that it takes force to change the direction something is moving in.  If you don't have a dolly with rotatable wheels, you can probably improvise something along these lines.
Once you've built up that framework, you can show him the same principles apply to things spinning.  Dr. Motl's suggestion of a bicycle wheel is a good one.  First, note that it takes force to get the wheel spinning.  Then note that it takes force to stop it from spinning.  Then note that it takes force to change the direction of spinning (i.e. the direction of the angular momentum vector).  A bicycle wheel makes an excellent gyroscope, but may be hazardous for a 9-year old, so watch out.
Now you can try to explain the gyroscope using these two principles that the kid understands.  Stuff falls over because the earth's gravity acts on it with a force to pull it down.  For an object that's not spinning, all it has is the linear momentum resisting gravity.  When it's spinning, you have the linear momentum AND the angular momentum resisting gravity.  And that's why it takes longer to fall over.
Of course this is not a complete description by any means, and misses a lot of the extremely important details (like precession), but c'est la vie.  It's not as if you're gonna have the kid understanding all the details (like nutation) without going through the math.
A: Forgive me for answering my own question, but with some inspiration from the other answers here and some child like thought I believe I have found a great way to help my son gain an intuitive understanding of why spinning tops don't fall over. 
First of all, I asked my son to imagine a single marble contained within a box. 
I then asked him to guess what might happen to the marble if the box was pushed in a straight line from left to right. 
Then I asked him what would happen to the marble if the moving box suddenly stopped. 
This was to encourage him to think about the marbles apparent reluctance to start moving and reluctance to stop moving once it had started. I told him that scientists like to call this inertia. 
I then asked him to imagine a set of boxes, each containing a marble, attached to the circumference of a wheel. 
This time I asked him to guess what might happen to the marbles in each box when the wheel started spinning, when it remained spinning at a constant speed and finally when the wheel stopped spinning. 
This was to stimulate his thinking about the reluctance of the marbles to start moving, reluctance to stop moving once they had started and the marbles apparent desire to push outwards from the centre of the wheel. I told him that scientists like to call these phenomena rotational inertia and the centrifugal force. 
So, at this point I asked him to imagine what would happen to the marbles in the boxes if we tried to twist the rotating wheel about its axis. Twisting the wheel about its axis would effectively pull two sides of the wheel closer to the axis of rotation of the marbles. The key here is to appreciate the independence of the marbles axis of rotation from that of the boxes attached to the wheel. The outward force of the marbles away from their central axis of rotation would resist this change, thereby causing a resistance to the twisting action on the axis of the wheel. 
Now, thinking about individual atoms within a solid body as tiny little marbles locked within their own tiny little boxes made up of adjacent atoms gives him an intuitive way to appreciate why the spinning top does not fall over. No formulas required! 
A: I'm going to take a stab at this! Remember, this is trying to explain to a kid without using complicated maths or formal terminalogy.
First you need to explain how the speed of points on a spinning object changes with distance from the axis of rotation. Imagine (or better, draw!) looking down on the spinning top from above, when it is spinning perfectly balanced. It turns at a certain speed, a number of turns per second. Now imagine a dot on the outer edge of the spinning top, and another dot halfway toward the center. The two dots must do the same number of turns per second around the middle, but the outer dot travels in a bigger circle. It has further to go, so it must go faster in order to get all the way around in the same time.
Next, to explain how two different rotations create a third rotation. This is probably the hardest part to explain. Imagine a wheel mounted on a pole, like this diagram from wikipedia: 

http://upload.wikimedia.org/wikipedia/commons
The wheel is mounted vertically and can rotate around a horizontal axis, in addition the pole can spin the wheel around a vertical axis. Now imagine the pole is spinning the whole wheel around the vertical axis, similar to a spinning top. Imagine a point on the wheel near the outer edge (dm1 on this diagram) and imagine you rotate the wheel to bring this point toward the top. The way the wheel is spinning means the the circle the dot makes as it gets nearer the top must get smaller, so that part of the wheel has to slow down. It's going too fast for the smaller circle, so it creates a 'pull' on the wheel, shown by the arrow. Now imagine a point at the top of the wheel (dm2) and you rotate the wheel to bring it toward the outer edge. The circle made by this part of the wheel is getting bigger, so this part of the wheel has to speed up. It's going too slow for the bigger circle, so it creates a 'drag' on the wheel, shown by the second arrow. The way the wheel is being spun by the pole means that the left half and the right half of the wheel are moving in opposite directions, so a pull on one side and a drag on the other are in the same direction. 
Now you can show the same thing for points on the bottom half of the wheel, except that the pull and the drag on the bottom half are in the opposite direction to those on the top half. Pulling in different directions on the top and bottom of the wheel, makes the whole wheel want to flip over.  Now you have shown that spinning something in one direction and then rotating it in a second direction makes it want to flip over in a third direction, at a right angle to the other two.
Finally you can use this to explain why the spinning top can't fall over. The spinning top is spinning about its vertical axis, like the wheel being spun around by the pole. If the spinning top isn't perfectly balanced, if it leans slightly in one direction, gravity will try to pull it over in that direction. This rotates the top in that direction. Just like when the wheel is rotated while the pole is spinning, this creates pulls and drags on parts of the spinning top that make it want to flip over in the third direction, at 90 degrees. So when gravity makes the top lean over one way, the flipping effect tries to make it lean over in a different direction. The spinning top is pulled in both directions at once so it will actually lean over in a direction half way between the two (draw a diagram if it helps!). 
BUT! Here's the clever part! As soon as the top starts leaning in a different direction, that becomes the new direction that gravity tries to tip it over in. When the direction that gravity is trying to rotate the top in changes, the direction it tries to flip in also changes, to stay 90-degrees away. So the top is pulled in two new directions, and the direction it leans changes again. And the same thing happens, again and again... The result is that the direction in which the top leans constantly changes - it is pulled around in a circle! Every time gravity tries to pull the top over one way, the flipping effect makes it lean a different way. As long as the top keeps spinning, the flipping effect means it can't fall over!
A: Lets be practical and think with the mind of 9 year old, this is my approach:
Tell him to stretch his arms and start spinning and now tell him to fold his arms and start spinning again (make sure he doesnt do it vigorously ) and now ask his observation saying which one was easy and say the same feeling would have been to the spinning top if it could think , you can then show his feet (compare with the tops axial) and say you are able to spin without falling as you are maintaining your position and speed relative to this point 
Hope this will clear your child's doubt :)
A: Well, the angular momentum conservation is still the essence although it may be formulated in a different language.
The top is spinning around a vertical axis and the spinning around this axis can't disappear. if the top decided to fall, the spinning would either disappear or would be replaced by a totally different spinning around a horizontal axis, and Nature doesn't allow such a change of the amount of spinning to occur quickly. One has to have a torque to change the amount of spinning, some force attempting to change the rotation, but the torque acting on the bottom tip of the top is so small that with a fast enough initial spinning, it takes a lot of time to change the spin substantially.
Moreover, energy conservation guarantees that if there's no friction, the top can't ever fall.

More practically, I would probably take a wheel from a bicycle, made the kid hold it, rotate it quickly, and then make him or her feel the forces when he tries to change the direction of the wheel. This is a pretty nice yet simple toy in various science museums, including "Techmania" we have here in Pilsen. See also this page which contains the picture above as well as some other insightful games and experiments relevant for the angular momentum.
A: Faced by the same question and a background that includes courses in vector calculus, I have sought a simpler answer. 
My answer is much that same as to why one can easily balance on a typical bicycle. Bicycles are constructed so that the point where the extension of the front fork pivot would hit the ground is in front of the point where the front tire hits the ground; the distance is called the "rake".  When a forward moving bicycle rider starts falling to the right, the wheel turns such as to bring the bicycle underneath the falling rider and it the process restores his position. 
When a spinning top starts to tilt, the point in contact with the supporting surface shifts away from the axis causing the top to roll. The direction of the roll is such as to move the bottom of the top back underneath its center of gravity. Of course energy is used every time the top must lift itself, so it eventually slows down and crashes. 
A: Explaining conservation of angular momentum is a good idea. I would like to add another explanation that is probably at the 9-year-old level. Imagine the top from, well... the top. Let's say it is going clockwise.
Suppose the mass begins to tip to the right. In a short amount of time, the rightmost part of the top would have experienced a downward acceleration, but, at the same time, travelled, lets say, a quarter rotation, which would have moved it upwards again and towards us (with no gravity). If you add up these two independent movements, you find that the the vertical movements (turning up, falling down) have cancelled out, so that its net movement is more akin to rolling.
You might need to draw a picture or use a model to convey this.
While precession depends on the conservation of angular momentum, I think this explanation makes it clear what's going on kinematically, without relying on concepts that are hard to develop at that age.
A: Along time ago (1960's) I saw on TV a show where a scientist was explaining the working of a spinning top in front of an audience of children. He had a roundabout with a seat in the middle on which he sat a young boy. The boy held a metal bar with a wheel on its end (Quite heavy!)  The scientist spun the wheel and rotated the roundabout.
I cant remember what the mathematics was behind this but I knew the boy could easily move the heavy bar up and down. But most surprisingly the scientist put up a tick in front of the bar and roundabout stopped immediately. (Showing that the angular momentum was zero!)
I might not have recorded the details of the experiment accurately but maybe the BBC archives might have a recording or may be some one could replicate the experiment, record it, and post it on YouTube? 
