Why is t needed?

The process for calculating p-values is to start with the sample statistic, standardize it to get a test statistic, then transform it via a cumulative distribution function. In Chapter 1, that final transformation was denoted z, and the CDF transformation used the (standard normal) z-distribution. In the last video, the test statistic was denoted t, and the transformation used the t-distribution.

In which hypothesis testing scenario is a t-distribution needed instead of the z-distribution?