When is Chi-squared the most appropriate statistical test to use?

Chi-squared is about establishing whether the difference in results between groups is likely to be due to chance, or due to some genuine difference in the groups' properties. It aims to answer the question, "are these groups from the same population?"It may be different groups at the same time, or the same group at different times, but when the question asks about differences it is generally a giveaway that the Chi-squared test is most appropriate.