Round Brackets ( )
Round brackets (called 'parentheses' in American English) are mainly used to separate off information that isn’t essential to the meaning of the rest of the sentence. If you removed the bracketed material, the sentence would still make perfectly good sense.
EXAMPLE: Mount Everest (in the Himalayas) is the highest mountain in the world.
EXAMPLE: There are several books on the subject (see page 120).
EXAMPLE: He coined the term 'hypnotism' (from the Greek word hypnos, meaning 'sleep') and practised it frequently.
They can also be used to enclose a comment by the person writing:
EXAMPLE: He’d clearly had too much to drink (not that I blamed him).
Square Brackets [ ]
Square brackets (just called 'brackets' in American English) are mainly used to enclose words added by someone other than the original writer or speaker, typically in order to clarify the situation.
EXAMPLE: He [the police officer] can’t prove that they did it.
If round or square brackets are used at the end of a sentence, the full stop should be placed outside the second bracket, as the final punctuation:
EXAMPLE: They eventually decided to settle in the United States (Debbie's home).
EXAMPLE: Dante testified that it was the last time he saw them [the missing coins].
Oxford Living Dictionaries. 'parentheses and brackets'. Retrieved from https://en.oxforddictionaries.com/punctuation/parentheses-and-brackets.