Translation:These times demand the best of nations and peoples.
Using "of" would imply just the best people and nations, ignoring all the rest. Using "from" implies a reaction from all nations and peoples to do their best. Using the plural, "peoples" is rather common, implying distinct populations of people, not just a general mass of indistinguishable people. Perhaps this is like in the Southern USA people saying "you all", a very handy way of distinguishing whether you are speaking to one person or an entire group. Unfortunately, "you all" has become a term for one "you" or several or more "you" in ordinary use so sometimes one hears "all of you all" for the plural.
No, saying 'from the nations' would imply you're at a multi country summit, like the U.N.
'Of the nations', which should be 'Of nations' is like a speech, refers to countries in general. Perhaps a president telling his people that their particular country isn't the only one suffering since these difficult times gets the best 'Of nations'...
I get that they are not the same word, I'm asking if there is a difference in meaning. In US English, apparently there is a difference: http://www.infoplease.com/world/statistics/state-country-nation.html
In daily use, most people use the word "country," reserving "nation" for more formal uses, such as "The United Nations."
Does that same difference apply in Portuguese?
Why is "These moments demand the best of the nations and their people" incorrect? This should be acceptable since Portuguese sometimes uses "a" or "o" to indicate possession as in "a mãe coloca a filha a dormir cedo." It allows English speakers to use "people" which in infinitely more acceptable in English.
Is it wrong to say "the most" instead of "the best"? What would be a different translation for that? Also in English it's rather awkward to say "peoples," since "people" is plural by necessity. I know it's accepted both ways but I can't imagine an English speaker ever saying "peoples"...
No, it is not at all awkward to say "peoples". You missed the meaning here. A people is (as Paulenrique explains above) a group of individuals who share similar customs, maybe a language, etc.). Brazilians are a people, Italians are a people, Jews are a people. They are three peoples that may even live in different countries. They are three of the world's peoples.