What does it mean when physical theories are inconsistent?

Inconsistency between two theories just means that there are statements that one theory says are true, and the other says are false.

An easier example than the one you're asking about is the inconsistency between Newtonian mechanics and special relativity. Newtonian mechanics says that if you keep applying a force to a material object, it will eventually go faster than the speed of light, c. Special relativity says that this statement is false.

Your example of Newtonian mechanics versus Maxwell's equations is a lot more subtle. If you'd asked someone in 1890, they probably would have said that Maxwell's equations were consistent with Newtonian mechanics, but they simply described different aspects of nature. In order to maintain this consistency, they were forced to say that Maxwell's equations had their simplest form in one preferred frame of reference, which was believed to be the frame of the ether. What they didn't realize was that the transformation of distance and time measurements from one frame of reference to another was not described correctly by the equations they'd been assuming. Using the correct, relativistic transformations, Maxwell's equations have the same form in all frames. Today, physicists think of Maxwell's equations as being inherently based on special relativity; but that wasn't how people in 1890 thought of them.

I am hoping someone could explain, rather in-rigorously, the use of the word inconsistent.

Essentially, in this context, inconsistent means the two theories give different, incompatible answers to the same question.

In this specific case, the question is:

• If light (an electromagnetic wave) is measured to propagate at speed $c$ in an inertial frame of reference (IFR), what speed is measured in a relatively moving IFR?

For simplicity, assume the relative moving IFR has speed $v$ in the direction of the light beam.

According to Newtonian relativity, and the Galilean coordinate transformation, the light will have a measured speed of $c - v$ in the relatively moving reference frame.

Maxwell's equations unambiguously predict the speed with which electromagnetic waves propagate in vacuum is $c$

To reconcile this inconsistency - these different, incompatible answers to the same question - requires accepting one of three possibilities:

1. There is an ether - an absolute frame of reference in which the light propagates at speed $c$. This means that there is no relativity principle for electromagnetism. Experiments should be able to detect this absolute rest frame.
2. Maxwell's equations are incorrect and must be modified to give an answer in accordance with Newtonian relativity.
3. Newtonian mechanics is incorrect and must be modified to give an answer in accordance with Maxwell's equations.