Question

What's a chi-squared test, what's it used for (examples please), how is it done, and how does it work?

Answer 1

The Chi-Square test of independence is used to determine if there is a significant relationship between two nominal (categorical) variables.

Explanation:

That is, we use it to see if the observed correlation of data between two data sets is statistically significant. "Categorical" means that the data has been counted and divided into categories.

For example, say a researcher wants to examine the relationship between gender (male vs. female) and empathy (high vs. low). The chi-square test of independence can be used to examine this relationship.
http://www.statisticssolutions.com/non-parametric-analysis-chi-square/

Good description of the method application is here:
http://www.ling.upenn.edu/%7Eclight/chisquared.htm