This is a rather old topic, but I felt I might have what you're looking for.
In response to some of the answers, you write:
Since the angular acceleration is always tangential, I would expect that the top should spiral outwards until it falls to the ground.
Absolutely, that is what you should expect to happen. And it does ... momentarily. The final solution is a little more involved than just being uniform rotation around the vertical axis.
In order to understand this, imagine that you take a spinning top which you have just set down at time $t = t_0 $ on the ground. Now, what happens in the next instant is exactly what you intuitively expect - the top begins to fall under gravity's influence and $ \phi $ (see figure for notation) starts to increase going from $ \phi \rightarrow \phi + \delta \phi $ at time $t_1$. Consequently the angular momentum $ \mathbf{L} $ of the top changes.
This is similar to what happens in the 2nd figure on the hyperphysics page, where $\delta \mathbf{L}$ is in the direction of $ \delta \theta $, only now $ \delta \mathbf{L} $ in the direction $ \delta \phi $ and lies in the plane containing the longitudinal axis $L_A$ of the top and the central vertical axis $V_A$.
Increasing $\phi$ lowers the center of mass of the top and thus its potential energy by an amount $ -\delta U $. Assuming energy conservation, this translates to an increase in the kinetic energy $\delta K$ of the top. Since the top is constrained to have zero linear momentum, this $ \delta K$ contributes entirely to the top's rotational energy.
Keep in mind, however, that the top is now rotating around two different axes. One component is the original spinning motion around its own longitudinal axes and the other is the rotation induced by gravity around the direction $N_A$ normal to the plane containing $L_A$ and $V_A$. Therefore, the $\delta K$ must be appropriately portioned between these two motions. Let's see how this happens.
The moment of inertia of the top ($I_A$) around the axis $L_A$ is clearly less than that ($I_V$) around the axis $N_A$. This is true for all but the most oddly shaped tops. Convince yourself that this is the case. In circuits more current flows through paths with lower resistance. Likewise in mechanics more energy is transferred to the component with lesser inertia. Thus the greater portion of $\delta K$ will go to increasing the angular momentum of the top around its longitudinal axis $L_A$ by some amount $\delta L'$
Now, conservation of angular momentum requires that there be a torque corresponding to this increase. The effect of this induced torque is to cause the falling top to start swinging back upwards. In this way, instead of a spiral, the tip of the top traces out something like a cycloid as it precesses around the central axis.
However, the diagram seems to indicate that the top should be precessing in a circle, not a spiral.
The circular trajectory is an idealization only achieved in the limit that $\omega_s / \omega_p \rightarrow \infty$, where $\omega_s$ is the spin angular velocity and $\omega_p$ is the precession angular velocity. Any top with realistic values of $\omega_s$ and $\omega_p$ will have finite "wobble".
I would not have known of this rather elaborate dynamics if not for one of Feynman's lecture volumes (Part I, I think) where this question is considered in great detail!
The above write-up is a little on the hand-wavy side and there probably are errors in my reasoning. For the full kahuna look up the Feynman lectures !
Cheers,
The reason fluids flow off your hand while solids don't, is that fluids can change shape and solids can't. The molecules in a fluid want to stay together, but they don't care about the shape they're in, so gravity will cause them to spread out over your hand and flow off the sides. Solids can't change shape so they just stay on top off your hand, held in place by friction.
A bubble is a thin sphere of a water/soap mixture filled with air. The water/soap mixture has surface tension. This means that the molecules are pulling on each other to try and reduce the size of the bubble. But the air inside the bubble has air pressure. If the bubble gets smaller, the air pressure increases, pushing back on this thin layer of water and soap. This will result in a stable situation: the surface tension is pulling inwards, and the air pressure is pushing outwards, resulting in a specific size and shape. If the bubble somehow got smaller the air pressure would restore its size, and if it got bigger the surface tension would. If the bubble is deformed to something other then a sphere, the surface tension and air pressure are no longer regular and equal, and they will keep pulling and pushing until they are again, which, again, makes the bubble a sphere.
So in a sense, a bubble is behaving as if it was a solid, because it has a rigid shape and size. The bubble can't spread out over your hand and flow off the sides, because it wants to maintain its shape and size. And the bubble as a whole doesn't move as easily because of adhesion to your hand (the fluid-counterpart of friction). If you blow against the bubble or tilt your hand, the airflow or the gravity will overpower the adhesion, and the bubble as a whole will slide of your hand. It will never spread out and flow off unless you pop it, at which point there is no bubble to speak of any more, but just the water/soap mixture, which is a fluid.
In summary, a bubble has a somewhat rigid shape because of the combination of surface tension and air pressure. This means it can't flow, but only move as a whole. Adhesion between the bubble and your hand prevents the bubble from simply sliding off your hand.
I'm not great at this, but here's my attempt to phrase it as to be understandable for a child:
If something flows, it has to change shape. Fluids flow because they don't care about what shape they are. Solids, like a die, don't flow because they do want to be in a specific shape. A die is always a cube. Because of this, the die can only move as a whole. The die doesn't fall off your hand because there is friction between the die and your hand. Just like a piece of rubber, or a strip of anti-slip, on a table.
A bubble is a ball with air inside and a thin layer of water on the outside. Everything is made up of tiny things called 'molecules' (let's not get ahead of ourselves here). The molecules in a solid hold each other very tight, that's why solid things can't change shape. The molecules in a liquid pull on each other, but they don't hold each other. Because the molecules are pulling on each other, the water in the bubble wants to get smaller. But, the air inside the bubble also has molecules. Air is a gas. The molecules in a gas don't hold each other at all, they just wan't to get as far away from each other as possible. So the molecules in the air inside the bubble want the bubble to get bigger. If the molecules in the air are pushing harder than the molecules in the water are pulling, the bubble gets bigger. If the molecules in the water are pulling harder, then the bubble gets smaller. After a while, the bubble will become exactly so big that the molecules in the air are pushing just as hard as the molecules in the water are pulling.
Now if the bubble becomes smaller, the air molecules will push it out again. If the bubble becomes bigger, the water molecules will push it in again. So the bubble can't change shape. You can see this in a balloon (thanks to Bobson). Take an empty balloon. It is very small because the rubber is pulling the balloon together, and there is no air in the balloon to push it out. Now if you inflate the balloon, more and more air will get inside. So the air will push out harder and harder, making the balloon bigger. If you poke the balloon, you can feel the air pushing against your finger. And if you take your finger away again, the air pushes the balloon back into shape. This is exactly the same as in a bubble. Except the water will 'break' much easier then the rubber in the balloon. So you can't really poke it.
So just like the die, the bubble and the balloon want to be in a specific shape. This means the bubble can only move as a whole. The die couldn't slide off your hand because of friction. With the bubble something similar is happening:
Hold your hands in a cup and throw some water in. Now open your hands. The water flowed off your hands, but some of the water is still sticking to your hand. This is because the molecules in the water and the molecules in your hand are pulling on each other too. It's called adhesion. Because of this adhesion between the water at the bottom of the bubble and your hand, the bubble can't slide off your hand, just like the die.
Best Answer
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.