S2. Inertial and Non-Inertial frame of reference

"Absolute space, in its own nature, without relation to anything external, remains always similar and immovable.  Absolute, true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external." -Isaac Newton

Let us begin with the,

Newton's first law

1. if an object is at rest it will stay at rest if no force is acting on it, and
2. if an object is moving it will keep on moving at constant velocity in a straight line if no force is acting on it.

This law is also known as the law of inertia.

Now for the very simplistic view of the inertial frame of reference, we can simple say that the frame of reference where the Newton’s first law (i.e. law of inertia) is valid.

All this means is that when you perform an experiment in the stationary frame or the moving frame of constant velocity $v$, there should not any kind of pseudo force observed on the system being considered. Any cause of unexplained force in system under consideration in the train (moving) frame hints us that there must be some kind of net external force acting on the body.

                         

Thus we can say that, 

“Inertial frame of reference is the reference frame where the law of inertia holds..i.e in this fig. above the train frame(v) and the lab frame ”

and

“Non-inertial frame of reference is the frame where the law of inertia does not hold true. i.e. we experience some kind of pseudo force(fictitious force) in the system. ”

In conclusion, stationary frame and the constant velocity frame are the inertial frame, whereas the accelerating frame is a non-inertial frame of reference..

This video is a very good source for the very basic understanding of the inertial and non-inertial frame of reference.