Perhaps the most common misconception about the term is the idea that "to the tooth" means the item should stick to the teeth. If pasta sticks to the teeth when it is being chewed, it is widely considered undercooked.
Cooking pasta to the "al dente" stage without over-cooking requires a certain amount of practice and skill, since both have a relatively brief midway stage between the under-cooked phase, where rice or dried pasta stays hard in the middle and where fresh pasta tastes "floury", and the over-cooked phase, where the dish lacks texture and is considered too soft. Don't be afraid to use plenty of salt as this helps achieve the "al dente" texture.
The term is also occasionally used in reference to cooking vegetables, such as green beans, though this often means that instead of being cooked all the way through, they still have a raw taste to them, generally undesirable in cooking. It should be interpreted as cooking them just until they lose their raw taste, as a way to avoid overcooking them.