The feeling of mintiness - not technically a taste, according those who are in charge of deciding on such things - is a result of mint's local anaesthetic action; it numbs your temperature sensors, giving you a cool feeling. It also numbs your taste buds, which is one of the main reasons it is used in almost all toothpastes (otherwise they would taste rank) and is also the reason why everything tastes weird when you've just eaten a mint.

In the days before proper anaesthetics were invented mint was often used in this capacity. So were cloves (either whole or as clove oil); these have a somewhat similar, but more powerful numbing effect.