(written from a Production point of view)
Referred to in the second revised final draft script of TOS: "The Corbomite Maneuver" (dated 20 May 1966), the Sino-Western trouble was a conflict on Earth that Uhura described as "World War III, almost." Captain James T. Kirk commented about the conflict, "The world was lucky it was stopped in time. None of us here would be enjoying life today if..." though he then trailed off.
Earlier in the episode's development, NBC executive Stanley Robertson wrote about this conflict in a letter to Gene Roddenberry (it was dated 17 May 1966 and concerned the first revised final draft of the episode's script). In the letter, Robertson stated, "As we discussed, a suggestion would be that on Page 48 of this draft, it be made pointedly clear that there was no 'World War III' between the Sino-Western powers. Let’s keep emphasizing with our writers, as we know you have been, that this is a topic we’d like to avoid."