A non-inertial frame of reference does not have a constant velocity. It is accelerating. There are several ways to imagine this motion:
Such an accelerating frame of reference is called a non-inertial frame because the law of inertia does not hold in it. That is, an object whose position is judged from this frame will seem to spontaneously change its velocity with no apparent non-zero net force acting upon it. This completely violates the law of inertia and Newton's laws of motion, since these laws claim that the only way an object can change its velocity is if an actual non-zero net force is applied to the object. Objects just do not start to move about here and there all on their own.
This is really quite easy to understand. If you are in an automobile when the brakes are abruptly applied, then you will feel pushed toward the front of the car. You may actually have to extend you arms to prevent yourself from going forward toward the dashboard. However, there is really no force pushing you forward. The car, since it is slowing down, is an accelerating, or non-inertial, frame of reference, and the law of inertia no longer holds if we use this non-inertial frame to judge your motion.
If all of this is viewed relative to the ground, it becomes clear that no force is pushing you forward when the brakes are applied. The ground is stationary and, therefore, is an inertial frame. Relative to the ground, when the brakes are applied, you continue with your forward motion, just like you should according to Newton's first law of motion. The situation is this: the car is stopping, you are not; so, you head out toward the dashboard. From your point of view in the car it seems like you have spontaneously been pushed forward. Actually, there is no force acting on you. The imagined force toward the front of the car is a fictitious force.
A similar fictitious force can be noticed by a person in a car when it speeds up. Let us say that you are in a car at a stop light. The car is standing still. The light turns green, and the car accelerates forward. While undergoing this acceleration, the car is a non-inertial frame of reference. If the acceleration is large enough, you will feel yourself "pushed" into the seat. Actually, no force is pushing on you. Again, as viewed from the inertial frame of the ground, you are just maintaining your velocity, as you should according to Newton's first law of motion. You were still when the light was red, and you are attempting to remain still when the light turns green. However, the car started to move when the light turned green. The car actually comes up from behind you, and, using the seat, the car pushes you forward. As the seat comes forward and pushes on you, the back seat cushion compresses a bit. You may interpret this feeling as your body being pushed backward into the seat. Really, you are attempting to maintain your velocity of zero, and the seat is coming up from behind to push on you. There is no backward force. The imagined force is a fictitious force. Fictitious forces arise in non-inertial, or accelerating, frames of reference.
There are several ways to describe a non-inertial frame. Here are a few descriptions:
What follows here are two demonstrations that show non-inertial frames of reference. The first one is an animation of a non-inertial frame which acts like an elevator. The other shows an animation of a rotating frame of reference. Rotating frames of reference are non-inertial frames since they are following curved paths. Remember that a change in direction, which would occur along a curved path, constitutes a change in velocity, and, therefore, constitutes an acceleration. If the frame accelerates, it is a non-inertial frame.
Elevator Example of a Non-inertial Frame