This page summarizes covariance and correlation formulas in both scalar and matrix forms. Formulas to convert between the two are provided as well.
Note: correlation refers to Pearson correlation coefficient which is what most people refer to when talking about correlation and this is also the default in many softwares.
Covariance (scalar) is a number calculated from two variables (two vectors of same length). The same is true with correlation.
Covariance matrix of an Nxp matrix is a pxp matrix. Same is true with correlation matrix.
Scalar  Matrix  

Input  
Mean    
Standard Deviation    
Covariance  
Correlation  
Scalar  Matrix  

Correlation to covariance  
Covariance to correlation  
Correlation using covariance 
 "*" is matrix multiplication.
 Matrix multiplication is associative: (AB)C = A(BC)
Standard deviation and covariance are different for sample and population:
Sample  Population  

Standard Deviation  
Covariance  
Mean 
N = population size
n = sample size (n < N)
On the other hand, correlation value is the same for sample and population because their difference cancels out. When calculating correlation with any of the formulas with standard deviation in it, correlation formula will be different so the appropriate one should be used to yield the correct value:
Using s_{x}, s_{y}  Using σ_{x}, σ_{y}  

Correlation  
Similarly during conversion, if correlation is calculated by dividing standard deviation from population covariance, standard deviation has to be calculated with 1/N. If covariance is calculated by multiplying standard deviation with correlation, the result is population covariance if population standard deviation was used. The mean is always 1/n and not 1/(n1).
Sample is usually the default in software tools:
Excel  R  MATLAB  

Standard Deviation  Sample  STDEV.S() 
sd() 
std() 
Population  STDEV.P() 
  std(__,w=1) 

Covariance  Sample  COVARIANCE.S() 
cov() 
cov() 
Population  COVARIANCE.P() 
  cov(__,w=1) 

Correlation  CORREL() 
cor() 
corrcoef() 
The difference is small but important because, for example, if you did not know this, you would not be able to debug (check) if a system that uses these statistical methods is coded correctly.
If there is freedom to choose between sample or population standard deviation (or covariance), use population formula when quantifying variation of provided data only (e.g. standard deviation of exam scores). Use sample formula when estimating variation of overall population by analyzing a sample of that data. (e.g. standard deviation of heights of all humans on Earth calculated from 100 randomly selected people's heights) The latter is more common because you are usually trying to extrapolate and draw wider conclusions about the general population using a sample of that population.