Yesterday, I came across this sentence:
"I have attended different english language courses in the USA and Great Britain both."
Is this a correct use of the word "both"? If not, how should the sentence be structured?
Yes, it's fine. "Both" in your example is a floated quantifier -- the quantifier of a noun phrase is converted to an adverb, which winds up somewhere after the noun phrase. Details differ, but "all" and "each" can sometimes float, also.
I have attended different language courses in both the USA and Great Britain.