Brackets are a fairly uncommon punctuation mark. Their main use is in quotations: they can be used to clarify quotes. For example, say you want to quote the following passage:
“I finally got to meet Trent today. I had a really great time with him. He was a lot taller than expected, though.”
However, you only want to relay the fact that Trent was taller than the speaker expected him to be. In order to do this, you would write the following: “[Trent] was a lot taller than expected.”
The brackets let the reader know that while the word Trent wasn’t in the original quote, his name was implied there. When using brackets, you need to be careful not to change the original meaning of the quote.
Another use of brackets is when there is a spelling or informational error in the original quote. For example, “Gabriel sat down on the river bank to fed [sic] the ducks.” (The term sic means that the typo was in the original source of this quote.)