Meaning of covariance Can someone please give me an intuitive explanation for the meaning of covariance between two random variables? What does it measure?!
 A: Covariance measures the direction and magnitude of the relationship between two variables $X$ and $Y$.


*

*Covariance is large and positive when $X$ and $Y$ are both large and positive at the same time. The bigger the changes that occur in $X$ and $Y$ in the same direction, the bigger the covariance gets. 

*Covariance is large and negative when $X$ is large and negative at the same as $Y$ is large and positive, and $X$ is large and positive at the same time as $Y$ is large and negative. The bigger the changes that occur in $X$ and $Y$ in the opposite direction, the more negative the covariance gets.

*Covariance is close to zero when there is no relationship between $X$ and $Y$, or when $X$ and $Y$ are small.


Covariance is related to correlation. Correlation is covariance "with the magnitudes taken out": the correlation between $X$ and $Y$ is the covariance between $X$ and $Y$ divided by the product of their standard deviations. Whilst the range of covariance is unlimited, the range of correlation is limited to $[-1,1]$, but it takes values close to $-1$ or $+1$ when $X$ and $Y$ are related even if $X$ and $Y$ are small.
Note that for both covariance and correlation the relationship that is being measured is a linear relationship, and that two variables can be dependent and still have a low (or zero) covariance or correlation.
A: The covariance is not far from the Pearson correlation coefficient, since
$$\rho_{X,Y} = \frac{cov(X,Y)}{\sigma_X\sigma_Y}$$
so if you understand the correlation, you should understand the covariance.
As for the Pearson correlation coefficient, the covariance measure how much your random variables show a similar behavior. For example, if the covariance is high and positive, then "$X$ large" $ \leftrightarrow $ "$Y$ large" and "$X$ small" $ \leftrightarrow$ "$Y$ small". 
If the covariance is negative, then the variables show opposite behavior : "$X$ large" $ \leftrightarrow $ "$Y$ small"
If the covariance is close to zero, then the variables tend to behave separately. If $X$ and $Y$ are independent, then the covariance is null (it's necessary but not sufficient, be careful).
While the Pearson correlation coefficient is between $-1$ and $1$, the covariance is between $-\sigma_X\sigma_Y$ and $\sigma_X\sigma_Y$. So if someone tell you "I have a covariance of 3 with my variables", it doesn't tell you anything, since you don't know the original standard deviation of the variables, you can't say if the covariance is high or not.
You can also take a look at the Wikipedia page : http://en.wikipedia.org/wiki/Covariance, it helped me a lot to understand basic concepts like this.
