How does acceleration affect motion?
Acceleration has two definitions:
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.