Velocity and speed are quantities which we have a good intuitive feel for, but often over-complicate. We know very well what the speedometer in our car measures: how fast we are moving. That is the distance we're traveling in a given time. That is all speed is, the rate at which you're moving. So if you want to calculate an object's speed you need only look at the distance it traveled and the time it took to travel that distance. Dividing the distance traveled by the time gives the speed.
Once we have the concept speed there is a natural question: where is the object going? To answer this question we need something more specific than just the speed. This is velocity. Velocity is a a vector quantity (i. e. it has a direction), which tells us not only how fast an object is moving, but also what direction it is moving in. Is it moving 30mph North or 30mph West? Physically these two situations should be distinguishable, so we need the more detailed information which velocity provides. Speed then is just the magnitude of the velocity vector (this only true for the instantaneous speed and velocity, see below) , neglecting the direction information.
Now that we've talked about speed and velocity once you may want to address a finer point and look at the question on average and instantaneous velocity© 2008 Kansas State University Physics Education Research Group.