The problem with dictionary definitions is that while all options are covered, it's unclear which is are most commonly understood. In general, it's Scottimandog's first definition, a vital or pivotal point. The whole issue a situation 'balances' on. We use it in a larger phrase, such as "the crux of the matter", "the crux of the issue" or "the crux of the problem". The crossed paths in the situation (the crux!) also mean there is a difficulty that needs to be solved; things are crossing paths and clashing.

[kruhks] –noun,pluralcrux·es, cru·ces  /ˈkrusiz/ Show Spelled[kroo-seez] 1.a vital, basic, decisive, or pivotal point: The crux of the trial was his whereabouts at the time of the murder. 2.a cross. 3.something that torments by its puzzling nature; a perplexing difficulty.