Correlation refers to a measure of how strongly two or more variables are related to each other.  

A positive correlation means that high values of one variable are associated with high values of the other.  Or if you like, the variables increase together.

A negative correlation means that high values of one variable are associated with low values of the other.  Or if you like as one variable increases the other decreases.  Note that like a positive correlation, a negative correlation still indicates that some kind of relationship exists. 

If there is no correlation between two variables they are said to be uncorrelated. 

Don't let yourself fall into the trap of believing that when there is a strong correlation between two variables that one of the variables causes the other.   Association does not mean causation.  For example, there is almost certainly a very high positive correlation between the length of peoples right arm and the length of their left arm.  But the length of a persons left arm did not determine the length of their right arm.  They are both determined by other factors i.e. genetics, diet etc. 

A correlation coefficient refers to a number between -1 and +1 and states how strong a correlation is.  If the number is close to +1 then there is a positive correlation.  It the number is close to -1 then there is a negative correlation.  If the number is close to 0 then the variables are uncorrelated.

Correlation coefficient can be calculated in a number of ways such as with a Spearman Rho. 

Correlations are very good for showing possible relationships between variables and some times are the only practical or ethical way of carrying out an investigation.  However, they cannot demonstrate a cause and effect.