This page is on the power-up. For the character, see here.
When wearing it, Mario becomes Tanooki Mario. Like the Super Leaf power-up, after a running start, this power-up allows Mario to jump up and fly/glide in the air by swing his tail for a certain period of time. The Tanooki Suit differs in its unique ability to transform Mario into a statue for a few seconds, during which he is invincible but unable to move. In statue form, he can destroy some otherwise invincible enemies by landing on them. He can also land on spikes without getting damaged.
If Mario uses a P-Wing while wearing the Tanooki Suit, he keeps the suit while gaining the permanent flight ability of the wing.
This power-up is most certainly inspired by the Japanese tanuki, a raccoon dog that, according to Japanese mythology, has mischievous behavior and shapeshifting powers.
The Tanooki Suit reappears in Super Mario 3D Land. Mario can attack with the tail and also hover after a jump but is unable to fly in-game. When Luigi gets the Tanooki leaf in this, he turns into Kitsume Luigi, based on the Japanese name for a fox "Kitsune", as opposed to Tanooki Luigi. Kitsune Luigi is essentially the same as Tanooki Mario, aside from looking like a fox. Tanooki Suit power-up item is absent from the game, and the Super Leaf grants the Tanooki suit instead. The same applies to Super Mario 3D World.