Question: I've been doing physics problems regarding cars for a while. I understand that there is a static friction (which appears when the wheel is rolling) and kinetic friction (which appears when the wheel is sliding). However, the way I'm visualizing it, static friction between the tire and road should not stop a car. In fact, when I asked this question to my teacher a long time ago, he said that it is actually the friction between the axle and the wheel that stops the car, and the road friction actually helps the car to move. But I know that when the car slips, the friction decreases and thus stopping time increases. How could this possible be linked to the axle? What is going on?

Answer: 
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> the way I'm visualizing it, static friction between the tire and road should not stop a car.
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Static friction is able to supply a force. If that force is opposite the direction of motion, it is able to stop the car. In the case of your car and the brakes, that's exactly what happens.

A torque from the brakes is applied to the wheel. This torque becomes a force against the ground. As long as the force is not too great, the wheel doesn't slip and the road supplies a force back on the wheel (which slows the vehicle).

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> I don't think the vehicle is slowed down by the static friction between the wheel and road. 
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And yet it is. To see that this is true, let's imagine a situation where we remove static friction. Drive the car onto a patch of ice where we assume the coefficient of friction drops to zero. The car continues to drive at the same speed.

When we stomp on the brakes now, a torque is still applied to the wheel, but now the force of static friction is zero. The car does not slow down and continues at the same speed. Only when (hopefully static) friction is present can we slow the car.

Kinetic friction is also capable of slowing the car, but we don't want that because we don't want to skid the tires.

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> ...that should be the static friction between the car and the road
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The only portion of the car that touches the road is the wheels/tires. That is the only part where friction can develop. So to say the friction between the car and the road or the friction between the wheels and the road is the same thing.
cars stop with their brakes, which produce friction at the inside of the wheel assemblies as the wheels rotate. This friction force retards the rotation of the wheels and dissipates the kinetic energy of the car into heat in the brake parts. 

since the wheels are in rolling contact with the pavement, and since the wheels are being slowed by the brakes, the pavement is pushing back against the wheels at their contact point with the pavement in a direction that opposes the movement of the car. So the car as a whole slows down.