Q1: I can't find the $\chi^2$-test (chi squared test) in my book. Is there an alternative name?
There aren't really alternative names. The chi-squared test is described in "statistics II for dummies".
The wikipedia article is already very fine though. I do not expect that the 'dummies' series does much better (disclaimer: I am not a fan of any "... for dummies" book... they teach you to remain a dummy).
Q2: What should I look for? Part I
statistics for dummies
The first book from the series 'statistics for dummies' seems to speak about 2-way tables and 'checking independence' (chapter 19).
In this simple case (low number of categories) a test for equality two proportions is done (chapter 15).
A primer for data reduction
I can not find much online about this book. But I can find the questions in this book.
Question 10.5 in the chapter on 'tests of significance' is about the chi-squared test to determine independence. Also the term 'chi-squared distribution' occurs in the index of this book, which points to chapter 10.
Chapter 3 'structured tables' helps you to write and interpret tables. Making the inference more qualitative, e.g. the answer to question 3.3 speaks about 'interactions' without giving any calculated measure, number, or test, that supports it and basing it on an intuitive look at the numbers in the table.
Chapter 4 'observed' distributions' helps you to write and interpret frequency tables of (observational) sample distributions.
Q2: What should I look for? Part II
The chi-squared test is a test that can be applied to multiple types of problems. If you are strict then you could say that Pearson's chi-squared test is a bit less general and only applies to test in which the chi-squared statistic is calculated by the observed and expected frequencies or counts.
(there can be other statistics which are chi-squared distributed, e.g. the log of a likelihood ratio or the test statistic in the Cochran–Mantel–Haenszel test which is based on odds-ratios).
While I referred to the two-way table as an application that you may be looking for. You should note that the (Pearson's) chi-square test is more general than the application. So you should tell us a bit more information and clarify your question. Why, are you looking for the chi-squared test?
(historically the chi-squared test started as a goodness of fit developed by Pearson around 1900, with some creativity a contingency table and test of independence can be seen as a fit as well, fitting a model of Independence and then testing it's goodness of fit)