What does calculating the slope of a graph line tell you about the data

The slope of a line is a characterization of it's 'inclination'. A line with a high slope is very steep, while one that has a very low slope is almost horizontal. A negative slope would mean the line is going down from left to right (in a coordinate system) and a positive slope would mean going up. 
In correlations, when a line is fit, a positive slope would mean a direct correlation where in an increase in one variable corresponds to a proportional increase in another; whereas a negative slope would indicate an inverse correlation.
In mathematics, slope is usually defined as 'rise over run' - or the ratio between the change in the y-variable and the change in the x-variable (m = (\Deltay) / (\Deltax) ). As alluded to in the first two paragraphs, a slope can be used to determine the relationship between two variables. If the slope is high, that means a tiny change in the variable x results to a huge change in the variable y. This could mean that y is very sensitive to x. The opposite is true if the slope is low. For instance, if the slope is one, a corresponding change in x translates to an equal change in y. For example, if each student is required to buy a textbook, then if a class size increases from 30 to 40, then the number of books purchased would also increase by 10 (since each student will get a book). 
Meanwhile, the sign of the slope represents how the two variables are related - either direct or inverse. For instance, an increase in the cost of a certain product might result to a decrease in sales. While this may not necessarily be a line, if a line is fit to the data, we would expect it to have a negative slope.
Hence, calculating the slope of a line fit to a given set of data will reveal information including how the variables are related and how strong this relationship is.
http://mathworld.wolfram.com/Slope.html