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.