Newton stated that an object at rest will stay at rest, and one in motion will stay in motion (the concept of inertia). To change this requires force. Applying force will result in a change of velocity (speed with direction) over the given time interval that the force is applied. This is acceleration.
One can accelerate in any direction, although from our personal experiences driving, we usually think of accelerating in the direction our car is pointing. However, changing direction is an acceleration. So, is changing speed without a change in direction.
You can sense these forces when you apply the brakes in your car, or when you turn your car onto a side street without stopping.
It helps to consider the force that is generating the change in velocity when thinking of acceleration. It makes sense that acceleration starts when the force is applied and ends when the force is withdrawn. That becomes the time interval during which velocity changes.