Tonsil stones are also called tonsilloliths. The tonsils are made of deep crevices. Sometimes food or debris can get stuck in the crevices, and over time build up into a "stone". The tonsil stones are firm, and may have a cheese-like consistency, and smell. They can cause throat irritation or bad breath, but sometimes cause no problems at all! They are very common.
To try and remove them you can use gargles. This would physically disrupt them and dislodge them, so that your child can then spit them out. Make sure that your child understands gargling before you try though! What you gargle with probably doesn't matter too much, it is the physical action of gargling that does the trick. Salty water is likely to be as effective as an expensive gargle.
Tonsil surgery can be helpful if the tonsil stones are a problem and gargles are not effective. However, tonsil surgery for tonsil stones is not funded by the NHS in most parts of the country.