Speed versus velocity
Your car's speedometer shows speed — how fast you are going, with no direction attached. That makes speed a scalar. Velocity is speed plus direction, so it is a vector. Two cars both reading 60 km/h but driving toward each other have the same speed and opposite velocities, and that difference is exactly why they can collide.
Average velocity: displacement over time
Average velocity over a time interval is simply how much the position changed divided by how long it took. Because displacement carries a sign, so does the average velocity — a negative value just means motion in the negative direction.
Average velocity is the change in position divided by the elapsed time. Its SI unit is metres per second (m/s).
For example, if a runner moves from x_i = 0 to x_f = 100\text{ m} due east in 20\text{ s}, the average velocity is 100 / 20 = 5\text{ m/s} east. Notice this says nothing about whether she ran evenly or sprinted the last stretch — an average deliberately smooths over the details.
Instantaneous velocity: zooming in
To recover the details, shrink the time interval. As \Delta t gets smaller and smaller, the average velocity over that shrinking window closes in on the velocity at a single instant — the instantaneous velocity, which is what a speedometer is really trying to show you.
Instantaneous velocity is the limit of average velocity as the interval shrinks to zero — the derivative of position with respect to time.
Reading motion off a graph
The position–time graph is one of the most powerful tools in kinematics because its slope is the velocity. A straight line means constant velocity; a horizontal line means the object is at rest; a line sloping downward means it is moving in the negative direction; and a curve means the velocity itself is changing — which is exactly the doorway to acceleration.