There are two types of friction: static and kinetic.
Static friction tries to keep two things that aren't moving so that they aren't moving, while kinetic friction tries to take two things that are moving and slow them down so that they're not. This motion, of course, is relative.
The confusing thing about cars is that the car is moving relative to the ground, so shouldn't we be dealing with kinetic friction?
It's not actually the car that we're concerned with in terms of friction with the road, but its tires. Actually, a single point on each of its tires. This point is the point that is making contact with the road, which of course changes as the tire rotates.
When the tires are rolling, this is actually a static context, contrary to all intuition.
Consider a tank or a bulldozer: something with tracks. The end of each track has a semi-circle that is moving with respect to the ground, but but the long part of the track that is on the ground is stationary. The tank takes advantage of this static friction to propel itself forward. Now consider that we gradually shorten this long part that is stationary to the ground until it is only a single point. This is essentially what a tire is. (Actually, a small part of the tire will flatten out but for purposes of the model and all)
The static context's resistance to relative motion between the tire and the road is what allows the car to move in a circle. The tire will only rotate on one axis, and as it rotates on that axis, it moves in a different direction from the way the car wants to go. So the tire uses static friction to instead force the car to go the way it is pointed in exactly the same manner that it forces the car to accelerate when you push the gas pedal. And actually, this static context is what the car uses to decelerate when you push the brakes, too.
The kinetic context is what is better known as skidding. The coefficient of static friction will always* be higher than the coefficient of kinetic friction, which is why you accelerate and stop better when you're not skidding. Turning while skidding is even more unpredictable because rather than "enforcing" static contact, friction is now "coaxing" the car to go in the direction you want it to go, opposing the car's angular momentum.
How do the tires choose whether to roll or skid? It's a matter of the current state and the forces applied. Static friction and kinetic friction are governed by similar equations:
$$f_s = \mu_sF_n$$
$$f_k = \mu_kF_n$$
Where $\mu$ is coefficient of friction ($s$ and $k$ being static and kinetic, respectively). What the $f_s$ equation represents is the threshold for static friction. $f_s$ matches the lateral force applied but "maxes out" at $\mu_sF_n$. If you push something laterally with a greater force, it will swap to the kinetic context and begin to slide across the floor (or skid). At that point, the $f_k$ formula represents the force with which static friction opposes the sliding/skidding motion. The system will not return to a static context until the motion between the two bodies reaches $\vec 0$ (Note that this is different from the magnitudes of the motions being equal; they must also be in the same direction). This is why it's difficult to recover from a skid, and it's a good idea to let off the gas and steer against the skid.
*to my knowledge. If anyone can clarify or provide counterexamples, please do so.