An idiom (the quality of it being known as idiomaticness or idiomaticity) is a syntactical, grammatical, or phonological structure peculiar to a language that is actually realized, as opposed to possible but unrealized structures that could have developed to serve the same semantic functions but did not.
Monolingual native speakers in an insulated monolingual-native environment are mostly not conscious of idiomaticness (the quality or state of a construction matching the idiom of the given language), because in general their minds never reach for, or hear, other possible structures.
However, when adults study a foreign language, they become consciously aware of idiomaticness and the lack of it.
For example, in English it is idiomatic to use an indefinite article when describing a person's occupation (I am a plumber; she is an engineer), but in Spanish and many other languages it is not (soy plomero; ella es ingeniera), and a native speaker of English learning Spanish must encounter and accept that fact to become fluent.
The count sense of the word idiom, referring to a saying with a figurative meaning, is related to the present sense of the word by the arbitrariness and peculiarity aspects; the idiom "she is pulling my leg" (meaning "she is humorously misleading me") is idiomatic because it belongs, by convention, to the language, whether or not anyone can identify the original logic by which it was coined (arbitrariness), and regardless of whether it translates literally to any other language (peculiarity).