If a journalist is interviewing you and you blurt out something inappropriate, you will have to ask her not to quote you on that. Sometimes the word quote is used as shorthand for quotation, a passage of speech or writing that’s repeated word for word.
As a verb, to quote means to repeat someone’s words, attributing them to their originator. If you’re giving a speech on personal organization, you might want to quote Ben Franklin in it — he’s the master. When you write out a quote, you put the other person’s words in quotation marks (“Aha!”). Sometimes a price estimate is called a quote, like when a mechanic looks at your engine and gives you a quote for the cost of repair.