Three-way ANOVA, also called three-factor ANOVA, determines how a response is affected by three factors. For example, you might compare a response to drug vs. placebo in both men and women at two time points. Drug treatment is one factor,  gender is the other, and time is the third. Is the response affected by drug? By gender? By time? Are the three intertwined? These are the kinds of questions that three-way ANOVA answers.