Actually both your questions are related.
given r(t) r " (t) is the acceleration.
There are 2 factors in acceleration change in direction and change in speed.
The change in speed is simply d(|dr/dt|)/dt this is called the tangential component of acceleration denoted aT. If s is the distance traveled along the curve then ds/dt is the speed and d^2 s/dt^2 is the tangential component
The change in direction is related to the curvature k = |dtheta/ds|
where theta is the angle the tangent vector to the curve makes wrt
the horizontal. So curvature is the rate at which this angle changes with respect to the distance s traveled along the curve. If K is large the direction changes rapidly with respect to a small distance traveled along the curve.
The normal component of acc denoted aN is k(ds/dt)
You should soon study these in detail when you consider the arclength parameterization of a curve and the details will be filled in.