Tangential Acceleration: Tangential acceleration, a(t) = dv/dt. Its function is to increase the linear/tangential velocity and subsequently increase the angular velocity. Suppose you switch on a fan. In its initial motion, it speeds up from v = 0 to a certain value. This increase in speed, as well as angular speed, is due to tangential accelerational.
Normal Acceleration: Also known as radial acceleration or centripetal acceleration, normal acceleration a(n) = v2/r, where v is the tangential velocity. It is responsible to change the direction of velocity and thus keeping particle in circular motion.
