In the strictest mathematical sense, centrifugal force is a misnomer. Imagine we are standing on the earth, which is rotating (but we don't know that), and the moon is actually stationary in space (neglect gravity, say the moon is actually just a point in space). To us it will look like the moon is rotating around us. Of course we know that in circular motion, you must always have a constant inward force to keep the object in a circle, otherwise it would go flying away. In this case, a centripetal force term arises that plays that role, analogous to gravity in the real earth-moon system. [resisting urge to use math]. This term is necessary because otherwise the things we know about circular motion (namely that you need a force to keep it in a circle) wouldn't make sense to someone on the rotating earth- to them, it would look like it's being kept there by magic!
Just like we constructed a situation in which a fictitious centripetal force arises, we can construct a situation in which a fictitious centrifugal
force arises. First let's review a basic fact about circular motion: as an object moves around in a circle, at any instant if you "freeze" the system and look at the direction in which the object is actually moving, its velocity is tangent
to the circle. There is always some central force, and this force is constantly "turning" the object so that it does not continue in that tangent direction but rather stays in the circle. But if this force (such as a string) were suddenly cut, the object would continue along the tangent instead of continuing to move in a circle.
Knowing that, say we have a spinning tire-like space station in an otherwise empty universe. Say there is a person hanging on to the outside of the station, who lets go of a ball. The ball shoots out away from the station on a path tangent to where it was released. Now to us, viewing everything from a stationary point outside of the station, we are simply seeing the natural consequence of circular motion- the ball moves off in a straight line. Nothing weird about that. But what does the astronaut, who doesn't know the station is rotating, see? Well, he lets go of a ball and it flies outward from the point of release and curves to the side - centrifugal and coriolis forces. So once again, we see that these fictitious forces are simply ways of describing to a rotating observer things that are perfectly obvious to the stationary observer.
What does this have to do with a car in a turn? Well, just imagine that instead of releasing the ball on the outside of the station, the astronaut releases it on the interior by reaching towards the center of the station before letting go. Once again the ball immediately starts moving in a straight line from its point of release, but a few seconds later it crashes back into the interior walls of the station. The key word is "crashes" - because the ball was actually moving in a different direction (though not necessarily at a different speed) than the rotating station walls. This is exactly the force you feel when you execute a turn in a car. The car itself is now following the curve of a circle of some radius, but because you are not connected to the car very firmly, you were essentially "let go" when the car started to make the turn. So you move in a shorter straight line, while the car turns, and you "crash" back into the side. This is obviously a huge exaggeration of the movement, because it actually happens in one smooth, continuous motion. But still, I said "straight line" and that is at odds with the fact that you are thrown sideways as the car turns, not forward. If that's bothering you, this last little bit should clear things up.
Let's return to the interior of the space station. You've heard those proposals about using spinning space stations to simulate gravity, right? Well that's centrifugal force in action. It would really create "gravity" because it is a force that pushes things outward. But all of the explanations I've given have said that things only move in straight lines unless they are connected to the station, so how does that make sense? This is very hard to explain without drawing, but I'll try (might want to draw it out yourself... think of what things would look like from the perspective of the astronaut). Let's return to the astronaut releasing a ball on the interior of a spinning space station, except this time remember to keep track of where the astronaut is as the ball moves. We know that at the point of release, the astronaut and ball are moving at the same speed, but the ball takes a straight line path until it crashes back into the walls while the astronaut follows the curved path of the station walls. But really, if the station is large enough, the straight line path of the ball is pretty much the same distance as the curved path of the station wall. So both the ball and the astronaut start at the same speed, and travel the same distance... the ball stays right next to the astronaut the whole time! If he outstretches his arm and lets go of the ball, when it crashes into the wall of the station, the astronaut will be there waiting. It will be just like if you dropped a ball on earth and it hit the ground next to your feet. If the astronaut gives the ball a light toss towards the center of the station, it will again move in a straight line as the astronaut spins around the outside and eventually he is nearby at the point and time when the balls touches the wall again. What does that remind you of? Gravity! To the astronaut, the straight line motion will look almost exactly like gravity! How does this explain why you are thrown outwards in a turning car? Well, on the station the apparent acceleration is between the ball and the side of the station. If you took a ride on the ball it would look like you were being pushed away from the center of the station, towards the outer wall. So the relative acceleration is outward, even though in the stationary frame you are moving in a straight line. In the car, the body of the car is the space station and you are the ball, so you feel like you are being pushed out from the center of rotation, which is why you get thrown sideways relative to the car.
In the big picture however, you are still moving (or at least trying to move) in a straight line.
There's one other interesting consequence if you think about the spinning space station. If the space station is relatively small compared to the humans inside, the approximation I made earlier about how the straight line path of the ball is about the same distance as the curved path of the astronaut isn't very accurate. In this case, the straight line is dramatically shorter, but initial velocity of both the ball and astronaut is the same upon release. What does this mean? Mainly that the ball will never land back where the astronaut is standing, it will always land "ahead" of him. If you throw the ball hard enough straight up, you will never be able to catch it again without moving! Or, you would have to throw it "behind" you (opposite to the direction in which the station is rotating) in order to slow it down a bit, in which case you'd be able to catch it (if you got the speed and angle exactly right). Because of these strange effects, centrifugal force is not an accurate simulation of gravity!
If you don't see what I mean let me know and I can find or draw pictures to show what I'm trying to say in words.
As a final note, you should be able to see now why centrifugal force is fictitious even though it causes very real effects. These effects are real and exist because you are touching a spinning body, and basically being thrown off of it, in a tangent line. To a stationary observer, no extra forces are needed to explain these motions, the traditional F=ma and knowledge of the conditions of central force circular motion are sufficient. But for someone who doesn't know he is on a spinning body, you would have to introduce extra force terms to make the motions they observe make sense. Otherwise they would see earth-like gravity occur on a space station with negligible mass and flip a shit.