A Z-test is a hypothesis test based on the Z-statistic, which follows the standard normal distribution under the null hypothesis.
The simplest Z-test is the 1-sample Z-test, which tests the mean of a normally distributed population with known variance. For example, the manager of a candy manufacturer wants to know whether the mean weight of a batch of candy boxes is equal to the target value of 10 ounces. From historical data, they know that the filling machine has a standard deviation of 0.5 ounces, so they use this value as the population standard deviation in a 1-sample Z-test.
You can also use Z-tests to determine whether predictor variables in probit analysis and logistic regression have a significant effect on the response. The null hypothesis states that the predictor is not significant.
You also have the option to use a Z-test to do a normal approximation for tests of Poisson rate and tests of proportions. These normal approximations are valid when the sample size and the number of events are adequately large.