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: 
Imagine a car just going along at constant speed. Do your free-body diagram. The net force on the car has to be zero. So the wheels have to be exerting on the ground (net at least) only a vertical force. 

Now imagine the car decreasing speed. There has to be a force opposite to the velocity. Free-body time again. The force on the ground has to include a force component that opposes the motion of the car. The wheels are the part touching the ground, so they must be supplying that force.

Remember your Newton's laws. To stop the car must be acted on by an external force. If you call the wheels "part of" the car, then the stopping force has to be applied by the ground. Meaning the wheels have to push back exactly as hard.

If you call the wheels "not part of" the car, then you can describe it as the braking mechanism applying a force to the wheels. Then it's "the wheel's problem" what it does with that force. But in that case, the car is stopped by friction between the brake mechanism and the wheel. It may be that your teacher is trying to get you to think that way.
Have you ever tried to lift yourself up by pulling your own hair? Go ahead try it. Try it really hard. You should be able to levitate a couple of inches, right? What? You got a handful of hairs in your hand and a bald patch in your skull? It should serve you right. Hopefully, you will learn that internal forces cannot change the total momentum.

The friction of the gears, engine, etc. are internal to the car. They cannot change the momentum of the car. For all we care, replace the gears, engine, and breaks with a couple elves and other mythical creatures (a.k.a. hidden variables). As long as those hidden variables are internal to the system (car) they cannot change the momentum of the car.

If you insist that friction cannot stop the car, I dare you to drive really fast on an icy road where there is almost no friction. Hooray for Darwinism.