Boiling stones are pieces of mineral put in a solution and heated in a round-bottomed flask so that boiling will be even. Without boiling stones, liquids heated in such flasks have a tendency to superheat without bubble formation and then violently boil all at once. This can be problematic in techniques such as distillation where one does not want superheated liquid pushed up into the condenser.
Boiling stones are also known as boiling chips, because chemists often use small chips of silicon carbide or calcium carbonate, The name "boiling stone" is a holdover from the early days of chemistry when experimenters put plain rocks in their solutions.
Boiling stones work by injecting an uneven surface area into a flask to promote bubble formation. This way a steady stream of tiny bubbles constantly and quietly forms as long as the liquid is at its boiling point. Forming bubbles requires nucleation, which is the "seeding" of a bubble, much like a crystal's formation on a surface. The inside of round-bottomed flasks is too smooth to be a good nucleation site, and lab chemicals tend to be free of impurities like dust that are effective for nucleation when boiling tap water on a stove.