What you're probably seeing is the dried up remains of the blossom, which sometimes adheres to the bottom of the developing tomato and resembles a tail. The dried flower is completely harmless and falls off with the slightest touch. Other vegetables, such as zucchini and squash, may also have "blossom tails" that remain on the ripening fruit. Gently remove the tail before you wash the tomato.
Catfacing is a term used to describe a physiological condition that causes tomatoes to become dimpled or puckered on the blossom end. In severe cases, the tomatoes may become very deformed or appear to have a tail. Holes in the tomato may develop necrotic, blackened areas. Cold weather is thought to be the main cause of catfacing. Avoid transplanting young tomato plants until at least two weeks after the last frost and use plastic cloches to protect them from cold nights.
Hybrid tomatoes are bred to produce uniform, predictable tomatoes, but heirlooms are another thing altogether. These robust plants grow on sprawling vines and produce rainbow-hued fruit in a variety of shapes and sizes. Purple Cherokee, for example, is a sweet, purple-fleshed fruit that occasionally develops deep rivets or bumps on the stem end that may resemble a tail.
A tail on a tomato usually isn't a sign of a serious problem. To ensure healthy tomatoes, though, place tomato plants in full sun, in rich, evenly moist soil. Avoid feeding tomatoes with nitrogen fertilizer until after the plants develop blossoms. Nitrogen fertilizer applied too early may cause catfacing or limited fruit production. Space tomatoes at least 18 inches apart so air circulates freely and watch for signs of pests or disease, such as yellowed, withered leaves and deformed fruit.