I’m having trouble getting my head around the difference between the different types of acceleration involved in circular motion.

So tangential acceleration $\displaystyle a_t$ is similar to the most common type of acceleration, such as a car accelerating in a straight line. Tangential acceleration is always tangent to the circle being traced by the objects motion.

Angular acceleration $\displaystyle \alpha$ is the derivative of the angular velocity with respect to time. So angular velocity is the rate at which an object completes the path around the circle and this means angular acceleration is how this is changing. I have no clue what direction this would be in.

Centripetal acceleration $\displaystyle a_c$ is always directed to the center of the circle. I don’t really see how this arises. Can this be measure directly?

Are all types of acceleration a vector?

for example, I would've thought that a satellite orbiting the Earth has tangential acceleration, but it has centripetal acceleration.