How can acceleration change the motion of an object?
It can change the direction or size of the velocity vector or both.
In most cases of an object which is accelerating the size (or magnitude) of the velocity is changed, so the object speeds up or slows down. The aspect that makes the difference is whether the acceleration is in the same direction as the velocity vector or in the opposite direction. If the acceleration is in the same direction as the velocity the size of the velocity will be increased (if it is in the opposite direction the size of the velocity will be decreased).
If the problem is one-dimensional then the acceleration only affects the size of the velocity vector and the direction will not be changed (so you only have to worry about whether the acceleration is in the same / opposite direction as the velocity). In two or three dimensional problems the size and direction of the velocity vector maybe change. Mathematical methods must be used to find the resultant velocity vector. An example of this would be projectile motion.
In some cases the acceleration is perpendicular to the velocity vector, in these cases the size of the velocity will not be affected. However the acceleration will change the direction of the velocity. A good example of this is circular motion with constant speed. During circular motion the velocity vector is always tangential to the circular path and the acceleration always acts towards the centre of the circle, so they are always perpendicular.
(Hence the speed remains constant because the size of the velocity vector does not change.)