Mini-turbo is a small, but fast boost you get after you finish drifting. Karts are allowed to perform a super-mini-turbo (Orange/red sparks), while bikes cannot (but they can perform a wheelie). The mini-turbo is only present whenever you're using Manual mode.
The higher the Mini-turbo stat is on a kart/bike, the more powerful and longer, the mini-turbo is. This stat becomes useless if you decide to go with automatic, allowing you to invest your points on other stats. (Using a different kart/bike)
The Mini-Turbo stat does not affect the colloquial "Stand-Still Mini-Turbo" which is the act of holding the gas and the drift button while stopped until blue sparks appear, then releasing the drift button. This can even be performed on Automatic by holding the gas and the brake while stopped. It is usually used as a recovery method by vehicles with bad acceleration such as the Flame Runner, as it is faster than accelerating normally.