What does it mean when I get a negative value when I do a definite integral?

The definite integral gives the value of the area bounded between the curve at the given limits, and the x-axis (if we are integrating with respect to x!) If this value is negative, it means that the area underneath the x-axis is greater than the area above the x-axis beween those two limits. Or it could just mean you've got your limits the wrong way round!