How do I determine if variables are correlated? Is it simply a mathematical calculation?
Yes, correlation is a mathematical concept, and it as well known as Pearson correlation. This is probably the one you are obtaining in your analysis.

However, there are other correlation analysis you can try in order to be totally sure about your result. The most famous ones out of Pearson are Kendall correlation and Spearman correlation.

In addition, sometimes relation between variables is less evident than a correlation analysis. For example, if the variables are time-related, one may have a lag over the other. For example, in finances it is known that the GDP is correlated with a lag of some months of the Employment Rate. It is part of a good Data Science knowledge to have a business knowledge, an intuition of why the variables may be related in some way and try to find how to prove that this non evident dependence exists.

Edit: I noticed you put R as a tag. You can study as well libraries polycor and ggm to learn about polychoric correlations and partial correlations, although the may not apply in your case.