In car driving, why does wheel slipping cause loss of control? When driving a car on ice, there is a danger of slipping, thereby losing control of the car.
I understand that slipping means that as the wheels rotate, their circumference covers a total distance larger than the actual distance traveled by the car. But why does that result in a loss of control? 
 A: Because friction is your method of steering! (- and of braking and accelerating.) As @MasonWheeler comments:

This is such an important principle that there's a special name for it: in the specific context of using applied friction to direct motion, friction is also known as traction.

Turning / steering
Friction is what makes you turn left at a corner: you turn the wheels which directs the friction the correct way. In fact, by turning your wheels you turn the direction of friction so that it has a sideways component. Friction then pushes your wheels gradually sideways and this results in the whole car turning.
Without friction you are unable to do this steering. No matter how you turn your wheels, no force will appear to push you sideways and cause a turn. Without friction the car is drifting randomly according to how the surface tilts, regardless of what you do and how the wheels are turned.
Braking and accelerating
Accelerating and braking (negative acceleration) requires something to push forward from or something to hold on to. That something is the road. And friction is the push and the pull. No friction means no pull or push, and braking and accelerating becomes impossible.

So, friction is very, very important in any kind of controlled motion of vehicles that are in touch with the ground. Even when ice skating, you'd have no chance if the ice was 100% smooth.
It should now be easy to grasp that it's a problem to go from static friction (no slipping of the tires) to kinetic friction (the tires slip and skid), simply because kinetic friction is lower than maximum static friction.
If you brake e.g., it is better to have static friction, because it can reach higher values than kinetic friction and thus it can stop you more effectively.
A: On ice, as the surface is so slippery, you may find that the tyre is always slipping to some extent. Because the tyre now rotates faster than needed for the speed of the car, a point on its circumference will cover a greater distance than the car. In other words, instead of being stationary with respect to the ground, the contact point moves.
Now how does this affect control of the car?
If you look at graphs of friction vs applied force, you'll see that friction increases with force, until movement occurs. Then it suddenly drops.
To explain this, you need to realise that, as you apply more force to an object and there is no movement of the object, the friction force has to increase with the applied force: if the two were not equal the object would start to move, in accordance with Newton's laws of movement. Once the force gets past a certain point, friction is no longer able to resist it and the object starts to move. At that point the friction force drops dramatically. 
You can understand why when you think of all the small imperfections of the 2 surfaces (the object and the substrate). These imperfections tend to mesh into each other, preventing the object from moving. Once the object is moving, the bumps no longer have time to fall into the depressions; instead they just skid over the top. Hence friction decreases.
What does all this have to do with a tyre? You need to realise that the point of contact between the tyre and the road is stationary with respect to the road. The contact point is different from moment to moment, but every such point is stationary on the road. It's only when the wheel locks up or slips that the contact point actually starts to move. At that point you loose traction.
Thus, on ice, because the contact point with the ice is not stationary, sliding friction decreases and you lose control of the car.
A: As all the answers have mentioned the reason for slippage is change of friction coefficient form higher to lower when going from static to dynamic friction. the reason for skidding and loss of control is two parts:  
The physics of motion and the driver's over steering.   
1- Tires have treads and indentations designed to impress the ice and make a shallow temporary microscopic grove to help traction and steering even in icy conditions.
They do this by slanted grooves on their treads which flex in a way to lead the car into turn smoothly. When the tire slips faster than the speed of car it grinds these imprints and skids off straight path and loses contact with the road. When the skid starts the suspension which was contracted under the dynamic loads gets free and expands suddenly in a jerk causing further instability and loss of authority of controls such as start of wild turns.
2- The driver not used to new low friction regiment thinks by over steering he could regain control but exacerbates the situation by plowing through any small imprint the tire has stablished and destroy weak traction starting to develop.  
The best way to regain the control is take advantage of car's momentum, let go of accelerator pedal gently and don't steer momentarily and let the car stablish a straight track, then when assured of traction steer gently and carefully.
A: While the handling difference has to do with differences between static and kinetic friction, it is not the difference between the static and kinetic coefficients of friction that explains loss of control, but the direction of the friction force.
As others have explained, in rolling without sliding the contact point between the road and the tire is stationary. In stationary friction, the friction force is directed in whatever direction is needed to prevent sliding from happening. When you turn the wheels of your car, movement forward happens in the rolling direction without the contact point sliding, and the static friction force is mostly directed perpendicular to the rolling direction, providing the centripetal force needed to make your car go around the corner.
But the moment the tires start slipping on the road, the contact point is no longer stationary. And when there is sliding at the contact point, the kinetic friction force is directed opposite the direction of sliding. Even if the total friction force stays the same, all of a sudden it is no longer providing the centripetal acceleration needed to go around the corner, but instead linearly decelerating your car. Which means you run off the road at a slower speed than you were cornering, but you still end up in a ditch.
A similar situation can happen when braking in a strong cross wind. While you are rolling without sliding, the stationary friction counters the sideways force of the wind to avoid sliding. But if you lock your wheels and start sliding, the kinetic friction force is applied against the direction of sliding, that is, mostly backwards, and without friction to oppose it the wind all of a sudden can push your car into another lane.
Or when trying to pull a cork from a bottle: if it is too hard to pull it straight out, you may find it easier to make it rotate. The moment it starts rotating, even the slightest force applied upwards will make the cork move out, because most of the friction is happening in the perpendicular direction.
A: The loss of control at the onset of skidding is due to the rapid change in the coefficient of friction between the tyre tread and the road from static to dynamic. 
The coefficient of static friction is larger than the coefficient of dynamic friction. 
This very rapid decrease in the frictional force requires the driver of the vehicle to react very rapidly especially when changing direction.  
The rule of thumb is after taking your foot of the accelerator and brake pedals steer in the direction in which the vehicle is travelling as this produces the maximum area of tyre tread with the road.
So if the car is skidding to the left steer to the left.
In general if the coefficient of friction between the tyre tread and the road is lower less force can be exerted by the road on the car and hence the driver's ability to change the direction and speed of the car is reduced.
A: In simple terms tyres need to deal with two types of force. The first being longitudinal forces (braking and acceleration) and the second lateral forces caused by cornering. 
The cornering forces are analogous to the tension in a weighted string swung around your head and are resisting the tendency of the car to continue in a straight line. 
The wheels on a car rotate easily in one direction (as they rotate on bearings)  but provide high friction in any other direction (you can easily push a car forwards but not sideways). 
I order to change the direction of a car you need to change its momentum and thus exert a force on it. This is achieved by turning the wheels so that you change their direction of free rotation relative to the direction of travel. This creates a reaction force at the contact surface between the tyre and the road which is transmitted via the side walls to the wheels, suspension and the main structure of the car. 
The total force that a tyre can support is limited and is a combination of the lateral and longitudinal forces acting on it so if you are braking or accelerating you are 'using up' some of the total friction available. 
SO once a tyre begins to slip there is suddenly much less difference in friction between rolling and sliding. 
Clearly once a wheel is actually spinning it has gone beyond this limit. In the case of rubber tyres the coefficient of sliding friction is much less than that of static (rolling) friction. 
If you spin the wheels when accelerating in a straight line you lose traction but not much else happens, however there is very little resistance to any lateral forces and so the spinning wheel can very easily slide. 
If the rear wheels break traction then overseer occurs and the rear wheels will slide away from the centre of the radius of the turn (towards the outside of the corner.) If not corrected immediately this can lead to a complete loss of control and a spin. 
If the front wheel lose traction the understeer occurs, the turning effect of the front wheels will be greatly reduced causing the car to continue straight on or at least follow a much larger radius turn than intended.  
To put it another way once the wheels begin to slide and thus reduce the friction to the road surface you no longer have any effective way of changing the momentum of the car. The crucial thing is that the coefficient of sliding friction of a tyre is much less than the coefficient of static friction. 
