The chi-squared test (χ²) is a statistical test for significance, e.g., for application to data from observations, to see if it gives significant support to a particular hypothesis or not.
Essentially, it consists of normalizing the data, summing the squares of the normalized data (the chi-squared), and using the resulting total to test for a degree of significance. Minimum chi-squared values have been calculated or can be calculated to indicate significance, from the number of categories in the hypothesis and the degree of significance one is looking for.
For example, if you hypothesize that most star-systems are binary, pick a hundred star-systems at random and check to see how many are binary, and wish to know whether these results indicate at least 95% chance that your hypothesis is true, you can calculate chi-squared and compare it with an appropriate value indicating 95% probability.
There are variants of the chi-squared test, but what is generally meant without other qualification is Pearson's chi-squared test.