Why doesn't a braking car move backwards? The net force on an object is equal to the mass times the acceleration, $F = ma$
When I brake on a (moving) car, the net force is negative, therefore causing the resulting acceleration to also be negative. This all makes sense, but if the acceleration of the car is negative, why does it not keep moving backward? I know cars in real life come to a stop, but I am having trouble explaining why the car does not continue to accelerate backward while the brakes are applied, with physics, so to speak. 
Where is the logic incorrect?
 A: In order to make it more "understandable", consider brakes that stop the wheels completely from turning. In this case, the car with the wheels blocked functions more like a brick.
Why doesn't a brick (or sled, or whatever else) start moving backward? Because friction forces only apply to moving surfaces.
When you have problems of understanding a complex system (like a car braking), try to simplify them - a car's braking system and its wheels add nothing to the problem or solution.
A: Friction is just an easy shorthand for describing a bunch of microscopic collisions between particles on the two surfaces.
When the wheel is spinning, bumps on its surface are colliding with bumps on the brake pad.
When the wheel stops, they are no longer colliding, because they aren't moving relative to each other.
It's the same as if you are standing in a strong wind.
The wind will push against you and accelerate you in the direction of the wind.  Once you are moving the same speed as the wind, it is no longer applying any force to you, because you are at rest with respect to the wind.
A: When you apply the brakes we all know they produce a net force backwards so we all know the resultant force is acting backwards. But when the resultant force acts backwards it does not mean that the object should travel backwards.
At the instant the brakes are applied, the object has a certain velocity, let's take it as $V_1$ which is positive, assuming the car moves backwards at the same instant, meaning it should have a $V_2$ which is negative. (Note that: velocity is a vector, so the direction is really important.) This situation causes the object to have an infinite deceleration, considering the resultant force stays constant.


*

*Deceleration can never be infinity.

*Once the object has a velocity, it will never travel backwards, because friction is an opposing force when there is motion. Now there is no motion, so no friction. Therefore no resultant force resulting in no motion.



Normally a graph for velocity when braking is like this.
A: When you say that the acceleration caused by braking a car is "negative", what you really mean is that its direction is opposite to the direction of motion.
If, in a certain instance, you mathematically model backwards motion as "negative motion", then in that case, you should actually say the acceleration caused by braking is positive. It is opposite in sign to the motion.
However, modeling motion and acceleration using positive or negative scalars only works if you are only interested in one dimension. More generally, you can model them as vectors, which are never "negative". Those vectors always have positive magnitude, but their directions are different.
In short, the fault in your reasoning is caused by getting confused over how to interpret the word "negative", and from what point of view a quantity can be considered "negative" or "positive". You are conflating the "negative" in "negative motion" and "negative acceleration", but those two expressions imply a different "reference direction" which is treated as "positive".
A: Cars move because the wheels are spinning in a certain direction. Brakes work by making the wheels not spin, not by making them spin in the opposite direction.
If instead of slamming the brakes you "brake" a car by having some other kind of force pushing  it backwards, like a super huge fan in front of it, then yes, it might begin moving backwards.
A: Friction as loss of energy rather than force
Two objects moving relative to each other have a relative kinetic energy proportional to the speed at which they are moving and the rotation they suffer.
When the surfaces of these objects are pressed together, friction occurs. This friction is just a process through which this relative kinetic energy is transformed into thermal energy, causing a temperature rise in both objects, and "sound energy" (the metallic screech that can be heard sometimes when braking), but this last part is negligible.
When applying this to the wheels of a car and its brakes, we see that the brakes and the wheels have no relative speed but, while the wheels rotate, the brakes do not move, as they are fixed to the car. This rotation only occurs when the car is advancing.
When you press the brake pedal, the system in your car presses the brakes against the braking surface in the wheels, creating friction and dissipating the kinetic energy of the wheels, which in turn slows down the car. As @senderle points out in his comment, some of the energy dissipated can be reused in cars, e.g. recharging the batteries. This happens until there is no more relative motion between the wheels and the brakes.
This means one of two things:


*

*The car has stopped and the wheels are at rest. No more relative motion so no more energy to dissipate. The car cannot go backwards because the car has no energy (movement means kinetic energy, and the car has none of it), and to make it move you have to provide it with energy, not take it away (in classical mechanics you cannot have negative kinetic energy, so 0 is the lowest energy state)

*The car is skidding (see @Eric Lippert's comment). This means that energy is dissipating only in the friction between the rubber of the wheels and the ground. The rate of dissipation in this case is incredibly lower that that of normal braking. This is the reason ABS is implemented in modern vehicles.
Note: In my country, when the wheels stop rotating but are sliding over the ground is called "wheel locking". I don't know how it is called in english, so if anyone knows please let me know so I can correct it if needed.
A: It's useful to think of this in terms of energy.
What the brakes are doing is converting some of the rotational kinetic energy of the wheels into thermal energy by frictionally heating the brake calipers.
This is why a lot of caliper design is focused on venting and cooling the hardware. Simply put, the reason the car doesn't move backward is that once the vehicle is stopped, there's no more kinetic energy in the system and the brakes have no mechanism to convert heat back into rotation. 
A: Brakes are applied to stop wheels from spinning gradually, whether it is backwards or forwards. We have to visualize all states of movements as separate instances or snapshots. When a car moves forward, brakes try to decelerate and stop car moving forward. As soon as the car enters in the state of moving backwards, brakes will try to stop moving it backwards too, deceleration in other direction. 
When the car is at zero speed, brakes are not really needed, until unless another force (wind/engine/collision/manual) is trying to push it backwards or forwards.
A: Acceleration has a direction, but its direction is not necessarily the direction of motion. It is the rate at which the velocity is increasing/decreasing. The direction of motion is the direction of the velocity.

Why the car does not continue to accelerate backward while the brakes are applied

It does accelerate backward, but again, if it is accelerating backward it doesn't have to be moving backward. It is the resultant force that is acting in the backward direction, causing the decrease in velocity.
A: The car does not keep moving backwards because the backwards force only exists while there is relative motion between the brakes and the wheels (the kinetic friction force). As soon as this motion stops, the force abruptly stops existing.
So, if the force persisted, you are right to think it would deaccelerate the car to a halt, and then keep accelerating it backwards. But this force is not an independent variable: it depends on the motion itself. The car stops, the force stops, and the resultant is no longer negative.
A: A notable property of frictional forces is that they resist motion (as opposed to other types of forces, which might resist displacement, for example, which is how a spring behaves). As a result, the brakes on your car slow down the motion of your wheels that produces forward movement of your car—but they also slow down motion that  produces reverse movement.
If instead you used another type of force-applying system to slow down your car (e.g., a giant spring), then your car would slow down, then stop, and then start moving backwards. 
A: Technically, you are making your wheels system interact with the brakes system. Causality is the result of interaction. Then, when the brakes system causes an action, the wheels system is expected to react (check the word: reaction). 
Brakes generate friction, and friction is always opposite to the movement, precisely because they generate a causal reaction during systemic interaction. This means that if you apply your brakes while climbing a mountain, the car will stop going forward. But when gravity is stronger than the motor force, the car will not go backwards, because the brakes will also oppose to the movement in such direction.
A: First of all, welcome aboard! I think that after all these answers it must be pretty clear to you now what's going on when a car brakes and comes to a full stop without going backward subsequently. But I just can't let the occasion go by to add one more answer.
I'm pretty sure you agree that when a book that moves over a horizontal surface the book will come to a full stop after some time (depending on how big the friction between the two is or, more technically, how big the coefficient of friction is) without reversing its motion after the stop. All the kinetic energy of the book is gone (mainly conversed into heat), and there is no more force left to make the book accelerate again. That is, if we assume that there is no other force than the friction force acting on the book, which is why the table must be horizontal because otherwise, a component of gravity gets a grip on the book.
Though it's not directly visible (which may have caused the confusion), the car comes to a full stop as a consequence of the same process by which the book comes to a full stop (which is obvious). The friction force between the braking blocks and the braking discs makes the car slow down. This is strange enough called braking by static friction. I write "strange enough" because static friction by itself can't move anything but this static friction ensures that the wheels can keep rotating without moving with respect to the road and hence the "kinetic" friction can do its work and make the care stop. 
If the friction between the braking blocks and the braking discs becomes too big the wheels will stop rotating and the road becomes so to speak the breaking block (I think you can understand why in this case the car won't reverse its motion, like in the case of the moving book on the table). This is called braking by kinetic friction and braking by static friction is just called like that to make a distinction between the two kinds of braking (of which the underlying mechanisms are the same). Because the distance to make a car stop becomes longer (if one can keep the car moving straight) in the case of braking by kinetic friction, and the car will become much more difficult to control (if you can't keep control, the distance to come to a stop probably gets shorter, but...) the ABS-system has a place in most modern cars.
I've read in one of the comments that after the stop it seems that the car moves a little backward. This effect is explained very well in this question.
