How does acceleration affect motion?

The two are equivalent, but generally imply the two sides of the discussion of "motion": where something is and where it is going.

Acceleration measures how something's velocity is changing. Whether we are discussing speeding up or turning, this is acceleration.

This also means it is another way that the object is moving (its second derivative). Hence, if you know that something is accelerating positively, you can imagine what the plot of position over time may look like, in a general sense.

Acceleration is the rate of change of an object's velocity. According to Newton's second law, acceleration is directly proportional to the net force acting on an object and inversely proportional to its mass. Acceleration affects motion by changing an object's velocity, either by increasing or decreasing its speed or by changing its direction. In the absence of acceleration, an object maintains a constant velocity.