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An inertial frame of reference is a frame that is moving at a constant velocity.
This means that the acceleration of the frame of reference is zero.
The most important law regarding inertial frames is Einstein's theory of special relativity, which states that the laws of physics must be the same in all inertial reference frames: there is no "correct" inertial reference frame.
This is a critical fact, and it means that, whenever one is solving a physics problem, one should feel free to use the inertial frame of reference that will make the problem the easiest.
A non-inertial frame of reference is just the opposite: it is a reference frame that is undergoing a nonzero acceleration.
Now, one will no longer see all of the same laws hold. For example, one must put in place fictitious forces to account for a reference frame that is constantly rotating, such as a person on a merry-go-round.
However, using a non-inertial reference frame can still be useful in the right scenarios.
For example, in the famous monkey-shooting problem, a hunter aims a gun directly at a monkey in a tree. At the same time that the hunter shoots, the monkey drops out of the tree. The result is that the bullet will always hit the monkey.
How do we go about proving this? Let us look from the poor monkey's perspective.
Since the monkey is falling out of the tree, it is accelerating downward with the gravitational acceleration .
But wait! The bullet is also accelerating downward with acceleration !
This means that, from the monkey's point of view, the bullet is not accelerating!
Since the bullet is moving in a straight line, and since the bullet was aimed directly at the monkey, the bullet will always be aiming directly at the monkey, and so it will eventually hit the monkey.
In an inertial frame of reference, we would have to pull out our and blah blah blah. This method was very quick!
One can see that non-inertial frames are still useful, but only in specific scenarios.