Tonsil stones, sometimes known as tonsilloliths, aren't food from last night's take-out; they're actually a combination of bacteria, dead cells and mucus that have gotten trapped in the pockets of your tonsils. By the way, you have pockets in your tonsils.
If you suffer from chronic tonsillitis, or have had chronic tonsil inflammation at some point in your life, you have an increased chance of having tonsil stones. People with dry mouth also seem to be more affected. Some people don't even notice, while others may feel discomfort when the debris hardens (which, if you can see it, looks like small white lumps of cauliflower in the back of your throat), including sore throat, swollen tonsils and ear pain. Because bacteria like to eat what tonsil stones are made up of, stones are also known to cause bad breath.
While you can remove the stones yourself with a swab or oral water irrigator and a mirror, or rinsing with an oxygenating or other non-alcohol-based mouthwash (it'll kill bacteria without contributing to dry mouth problems, like an alcohol-based product can), the only way we know to cure them is with a tonsillectomy.