Perhaps the most known slogan by the Donald Trump presidential campaign, 2016 is Make America great again.
Has he made any public statement exactly what period in history he considered that the United States was great?
I believe that this CNN article perfectly answers your question.
This New York Times article quoted him:
Asked when he thought American power had been at its peak, Mr. Trump reached back 116 years to the turn of the 20th century, the era of another unconventional Republican, Theodore Roosevelt, who ended up leaving the party. His favorite figures in American history, he said, include two generals, Douglas MacArthur and George S. Patton — though he said that, unlike MacArthur, he would not advocate using nuclear weapons except as a last resort.
The question is problematic, because Trump hasn't really articulated it that way. Any one dimensional approach that looks for a specific date when America allegedly used to be great, is trivial and misses Trump's overall message, which would be embodied in his various policy proposals, each concerned with rolling back something "new and bad" in favor of the "old and good", aka his own articulation of conservatism.
To put it another way, the phrase "Make America Great Again" is intentionally vague so as to cover all Trump's areas where he's pushed for reform, such as immigration and border security, ACA repeal, removing taboos on open conversations in Washington about Islamic terrorism, and exposing political corruption.
You asked for a specific example: in an speech in August Trump railed against the policies of the current administration that had allowed and enabled crimes committed by illegal immigrants. The specific aspect of "Mak[ing] America Great Again" that would apply to that speech would be repealing the Obama administration's immigration policies:
In the speech Trump discusses a list of people murdered by illegal immigrants, discusses other negative economic and other effects of illegal immigration, and expresses a desire to restore things to the way they were before the administration's immigration policies.
For another example, in the second debate Clinton claimed the leak from her well paid, private private speech to Goldman-Sachs about "public and private positions" was actually inspired by Abraham Lincoln. Trump castigated her both in the debate and later from the stump on that, alleging a contrast between "Honest Abe" and Clinton in terms of honesty and integrity. Drawing similar contrasts between alleged Clinton corruption and historical American statesmen regarded as virtuous is another theme Trump has repeatedly pushed, and obviously you can't nail those down to a single historical date, either.
This slogan was lifted verbatim from the first Reagan campaign in the 1980's. It has nothing to do with reality now, any more than it did then. At least, at the time of the Reagan campaign, there was a recession going on, Iran had 50 US hostages, unemployment was high and there was an oil crisis.
At the moment, although we have displaced workers in some geographic locations and industrial sectors, we have an unemployment index under 7% in most states, oil prices are near historic lows for the last 20 years, the stock market is still near record highs, and ISIS is in retreat in Iraq and Syria. I'd say we're pretty great right now, but my guess is that's not going to last.