Time rate of change of velocity = acceleration
If acceleration is constant in magnitude and direction, that means the force acting on a body is constant. Some thing like electromagnetic force in a uniform electromagnetic field. It is possible when one (net) force is acting on an object.
Gravitational force on a particle when no other acts - it is the case of a projectile motion in two dimensions. The projectile changes velocity in vertical direction and has a constant horizontal velocity. So its total velocity changes in magnitude and direction all the time. acceleration is always vertically downwards and constant at g.
In a motion along a straight line acceleration can be constant , but velocity does not change in direction at each instant. In a circular motion, acceleration or change of velocity - is not constant. it changes direction towards the radius always.
It is constant only in magnitude.