The gravitational time delay of light, also called the Shapiro time delay, is one of the four classical tests of Einstein's theory of general relativity. This article derives the Newtonian version of the Shapiro time delay from Einstein's principle of equivalence and the Newtonian description of gravity, in a manner that is accessible to undergraduate students and advanced high-school students. The derivation can be used as a pedagogical tool, similar to the way that simplified derivations of the gravitational deflection of light are used in teaching about general relativity without making use of the more advanced mathematical concepts. Next, we compare different general-relativistic derivations of the Shapiro time delay from the Schwarzschild metric, which leads to an instructive example for the challenges of formulating the post-Newtonian limit of Einstein's theory. The article also describes simple applications of the time delay formula to observations within our solar system, as well as to binary pulsars.