A bracket is a punctuation mark that's used to set a word or phrase aside from the rest of a sentence. Sometimes dates or other numbers in a sentence are enclosed by brackets.
When you quote someone in a piece of formal writing, you can use brackets around an ellipsis, or three dots, to show you've omitted some of the actual quote: "He exclaimed, 'What a surprise to see you here [...] looking so cheerful!'" Another kind of bracket is the L-shaped support that holds a shelf against a wall, often using screws. This is the original meaning of bracket — the punctuation mark was named for the way it resembles the carpentry support.