It was an intermittent process beginning in 1989 with the fall of the Berlin Wall, the fall of many Communist governments in Eastern Europe (especially Poland, Czechoslovakia, Romania, and, of course, East Germany). In 1990 several Soviet republics broke free of the Soviet Union and declared their independence and many republics held relatively free elections. By then it was clear that the USSR was on its way out and the power structures it had propped up (the Warsaw Pact, Moscow's hold over the other Soviet Republics) was gone. During 1991 the Warsaw Pact was officially dissolved as was the Soviet Union after an unsuccessful coup attempt. By the end of 1991 there was no longer the hint of doubt that the de facto state of an independent Eastern Europe and independent former Soviet republics would pertain indefinitely. Also during 1991 the US and the USSR (or what remained of it) signed START I, an extensive stand down in strategic nuclear weapons deployments.

I wouldn't say that 1985 would be a valid choice for the end of the Cold War, as the slight thaw in relations and the increase in openness was only the beginning of the end of the Soviet Union in hindsight.

One might choose any year between 1989 and 1991 as the end of the Cold War depending on how loose or demanding one was in terms of requirements.

I think the strongest case is made either for the year 1990, where many leaders who truly represented the people were elected to power in many Communist nations and the end of what amounted to a Soviet Empire had become quite apparent; or the year 1991, where the Cold War could be well and truly said to be over without nagging suspicions that there could be a reversal.