Non-inertial frames can be defined by defining inertial frames first. Then, every reference frame that is not inertial is, by definition, non-inertial.

The definition of inertial frames is based on Newton's first law.

An inertial frame of reference is a reference frame relative to which Newton's first law of motion is true.

To understand what this means we need to know Newton's first law. Here is a translation from the original Latin by by I.E. Cohen and A. Whitman.

Every body perseveres in its state of being at rest or of moving uniformly straight forward, except in so far as it is compelled to change its state by forces impressed.

To test whether a reference frame is inertial, take an object that has no forces acting on it and make sure that the object is at rest relative to this frame at one moment. If the object remains at rest relative to this frame, the frame is inertial.

In contrast, if an object stays at rest relative to some reference frame while there is a net force acting on the object, then this reference frame is non-inertial.

An example of a non-inertial frame is the frame of reference that is fixed relative to an accelerating car. Why is this a non-inertial frame? Let us consider a passenger in the car. The passenger is at rest relative to the car while experiencing a force that pushes him forward. (The back rest of his seat is exerting this forward force on him.) Therefore, the frame of reference moving with the car is not an inertial frame.

Any frame that is accelerating relative to an inertial frame is a non-inertial frame.