There's two common uses of this term - in philosophy of science and in mathematics (arguably it was also a philosophical issue when first described). u/o0oo0oo0oo0ooo's answer (as well as the Stanford philosophy page) are good resources on the philosophy of science one. Edit: It seems like that answer was erroneously removed - if you were actually asking about that, let me know and I can summarize it as well.

In mathematics, incommensurability is best understood through the classical Greek conception of numbers around Archimedes' time. The fundamental entities were the whole numbers 1, 2, 3, ... . There was no decimal notation, but it was obvious that numbers could measure lengths, and that lengths between the whole numbers existed. They developed a system equivalent to fractions, but they thought of them slightly different. If some interval had length 3/2, that meant that you could break it up into 3 intervals, each of which was the result of breaking up some reference interval into 2 parts. In general, fractions were not their own numbers, but could only refer to some kind of relationship between the numerator and denominator, which were numbers. Measuring a length using fractions actually meant you were comparing the lengths of various intervals, and those intervals are called commensurable - literally, co-measurable, or having a common measure.

This was ruined with the discovery of the 45-45-90 right triangle. If this triangle had legs with length 1, then its hypotenuse has length √2. As you may have learned, √2 is an irrational number, which means it cannot be written as the ratio of two whole numbers. This fundamentally broke the previous conception of the relationship between lengths, fractions, and numbers - we had found two lengths that were incommensurable, not able to be measured relative to each other (using the only kind of number that existed at the time, whole numbers).

They had discovered that their notion of length was broader than their numerical notion of measure. This was just a feature of classical geometry (some lengths were commensurable, some were not), but nowadays we don't make that distinction as much because people are comfortable using real numbers and Cartesian geometry, where irrational lengths are just another number. Also, nowadays instead of the above, we say that commensurable means expressible as a rational number, and incommensurable means irrational. This definition is identical, though it loses some of the historical context.

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