We know that velocity is a vector quantity which has a magnitude as well as a direction. So a change in velocity can be brought by either changing the direction or the magnitude. We are familiar with ways in which magnitude of velocity might be changed. A common example of this is slowing down of a rolling ball due to friction.
But can you think of an example in which the change in velocity is due to direction alone? If you are thinking of moving in a circle while keeping the speed constant, then you are right. In such a case, the velocity will keep on changing because of constant change in direction of motion, but the magnitude will remain the same. This is called uniform circular motion.
An experiment for uniform circular motion
Tie a small lump of clay to one end of the thread and hold the other end in your hand. Start moving the lump around in a horizontal circle. Now, if you release the thread you will see that the lump goes off in a straight line which is tangential to the circular path. The speed which the lump was moving around the circle is the same speed which the lump has as it flies off tangentially. This happens because of the fact that as the lump is released it continues to drift off along the direction in which it was moving at the moment. This confirms that the direction of motion changed at every point in the circular path of the clay lump. There are an infinite number of points in a circle and each of the those infinite points denote a direction which the lump would take if it was released.
For an object to keep moving in a circular motion, it is the centripetal force which keeps the object on the path. Centripetal force acts along a line joining the center of the circular path to the object. In the above example of moving a lump of clay tied to a thread, the centripetal force is provided by the tension in the thread. A fictitious force called centrifugal force acts in the direction opposite to that of the centripetal force and has the same magnitude. Centripetal force does not refer to a new kind of force. The centripetal force causes a change in the direction of the object without disturbing the speed. It is an unbalanced force and for an unbalanced force to not change the magnitude but only the direction is new.
Tangential acceleration is the acceleration in the tangential direction which causes a change in the magnitude of speed as well. For uniform circular motion, the tangential acceleration is zero because we do not want the speed to change. The net acceleration of the lump of clay at this point is calculated with the help of a combination of tangential acceleration and centripetal acceleration. Tangential acceleration is caused by a tangential force which is applied on the lump of clay.