Speed is defined as the ratio of distance traveled to time taken. That is,
Speed = distance/time
Velocity is defined as the ratio of displacement to time taken. In other words,
Velocity = displacement/time
The distance traveled can never be negative, as it is the path traveled to reach a location from an initial position. Think of distance traveled as the odometer reading of an automobile. Thus, an object cannot have a negative speed.
Velocity, on the other hand, can be negative. Since, velocity is a vector quantity (unlike speed, which is scalar), it has a direction component. Imagine, for example, we start from the origin and move in the positive x direction, stop after some distance and then head back in the -x direction, cross the origin, and stop after traveling some distance. Displacement in this case will be negative, since we use a traditional sign convention of positive sign for travel towards the right side. Similarly, the velocity of an object falling down is taken to be negative (down the y axis).
Thus, velocity can be negative, speed cannot be.
Hope this helps.