Covariance transforms a naturally abstract relationship between two variables into one that is easier to understand and predict; basically it proves that there exists a relationship between 2 or more variables/data sets (and determines the strength of that relationship). Taken directly it is the strength of the correlation between two variables. The correlation (found by determining the product of the average standard deviation of each variable from its mean) takes into account the unit differences between the variables being compared, scaling the relationship into a more precise measurement.
The correlation factor, which accounts for the magnitude difference between covariance and the product of the individual variances, is the ratio of the covariance and the product of the standard deviations of each. One way in which it differs from covariance is that it shows whether or not the relationship is positive or negative.
Just as standard deviation is related to variance covariance is related to the correlation. Mathematically covariance is the average of the product of the square root sum of the square differences between each variable and its mean within its set. With regards to economics the covariance is important since it helps determine risk by revealing the relationship between different stocks (for people who want to stay away from a specific stock or industry covariance can help them make better decisions).