Use brackets [ [ ] ] in the following situations:
You can use them to include explanatory words or phrases within quoted language:
Lew Perkins, the Director of Athletic Programs, said that Pumita Espinoza, the new soccer coach [at Notre Dame Academy] is going to be a real winner.
If you are quoting material and you've had to change the capitalization of a word or change a pronoun to make the material fit into your sentence, enclose that changed letter or word(s) within brackets:
Espinoza charged her former employer with "falsification of [her] coaching record."
See the description of the ellipsis for information on using brackets to set off an ellipsis that you have used to indicate omitted language in a quotation.
Also within quotations, you could enclose [sic] within brackets (we italicize but never underline the word sic and we do not italicize the brackets themselves) to show that misspelled words or inappropriately used words are not your own typos or blunders but are part of an accurately rendered quotation:
Reporters found three mispelings [sic] in the report.
(It is bad manners, however, to use this device to show that another writer is a lousy speller or otherwise unlettered. Also, use it only when it is important to maintain the original spelling for some reason. If you can edit (remove) the error without violating some scholarly or ethical principle, do so.) Note, also, that the word sic means "thus" or "that's how it was" and is not an abbreviation; thus, no period.
If you have italicized or underlined words within quoted language that was not italicized or underlined in the original, you can note that change in brackets included within the sentence or paragraph:
It was the atmosphere of the gym that thrilled Jacobs, not the eight championship banners hanging from the beams [italics added].
("Italics mine" or "emphasis added" would be other acceptable phrases.)
You can use brackets to include parenthetical material inside parenthetical material:
Chernwell was poet laureate of Bermuda (a largely honorary position [unpaid]) for ten years.
Be kind to your reader, however, and use this device sparingly.