Difference between position and velocity graphs?

Don't confuse acceleration, the change in velocity per unit time, with velocity, change in position per unit time.

Examples:
A car parked in the driveway:
- has a velocity of 0. It isn't accelerating. V(t) = 0.
- It also isn't moving. D(t) = 0.

Same car traveling on a straight road at a steady 100 km/h:
- has a velocity of 100 km/h. V(t) = 100.
- is changing position. D(t) = 100t.  Note: t is hours, D is kilometers.
- it isn't accelerating. a = 0.

Same car accelerating very slowly out of the driveway and down the road at a constant acceleration of 5 km/h^2.
- The velocity starts at 0, but increases at a constant rate. V(t) = 5t.
- During the first hour, the velocity of the car at any point in time is somewhere between 0 and 5. Every minute the car accelerates, it goes a little faster. Therefore, each minute, the car covers slightly more distance than it did during the previous minute.

Because the acceleration is constant we can compute the car's average velocity between time t1 and time t2. It is
  (V(t1) + V(t2)) / 2
The distance the car covers from time t1 to t2 is the average velocity times the elapsed time. So
  D(t1, t2) = [ (V(t1) + V(t2)) / 2 ] * (t2 - t1)
For simplicity, let t1 be 0.
  D(0, t2) = [ (V(0) + V(t2)) / 2 ] * (t2 - 0)
Simplify:
  D(t2) = [ (V(0) + V(t2)) / 2 ] * (t2)
Above, we said V(t) = 5t. Substituting
  D(t2) = [ (5*0 + 5 * t2) / 2 ] * (t2)
Simplify:
D(t2) = [ 2.5 * t2 ] * (t2)
D(t2) = 2.5 * t2 ^ 2 { a parabola }