Speed is the distance covered by a moving object in a single unit of time, for instance 1 second. On a distance-time graph, speed is the gradient of the graph, or the derivative of the equation of distance against time with respect to time.
Speed, however, has no direction associated with it. It is said to be a scalar quantity as it has zero dimensions. The speed of a moving object is calculated by measuring how far from a starting point A the object gets (say it gets to point B) in 1 unit of time. If A and B are say 3m apart, then we know that the object is moving at 3m per second say. To calculate speed, we only need to know how far A and B are apart in 3D (Euclidean) space (the space that we experience around us that is), or in 2D space if we are considering travel on the flat only. We do not need to know what direction B is from A.
Whereas, velocity has both direction and magnitude (as the character Vector says in Despicable Me), and not just magnitude as is the case with speed. It is said to be a vector quantity because it has more than zero dimensions. So velocity has a speed associated with it, measuring how far the object gets from point A (say it gets to point B) in 1 unit of time, but also has a direction associated with it. For velocity, unlike speed, we need to know what direction B is from A. For this, we could say how many degrees the direction of B from A is off North - if it is directly East of A, this would be 90 degrees clockwise off North; or, we could say what direction to go in in gridded 3D (or 2D if on the flat plane) space - say, to get to B from A go sideways 2, forwards 3 and upwards (towards the sky) 1, which we could write as a vector (2,3,1). Velocities are usually written in speed and vector form, so we would say the object is travelling at eg 3m per second in the direction (x,y,z) eg (x,y,z) = (2,3,1). The length of the vector determines the number of dimensions of space we are travelling in, which in the real spatial world is at most 3.