‘Sticks and stones may break my bones’ is a response to an insult, implying that “You might be hurt able to hurt me by physical force but not by insults”.

‘Sticks and stones may break my bones but names will never hurt me’ is a stock response to verbal bullying in school playgrounds throughout the English-speaking world. It sounds a little antiquated these days and has no doubt been superseded by more streetwise comebacks.

The earliest citation of it that I can find is from an American periodical with a largely black audience, The Christian Recorder, March 1862:

Remember the old adage, ‘Sticks and stones will break my bones, but words will never harm me’. True courage consists in doing what is right, despite the jeers and sneers of our companions.