The t-test and ANOVA are both considered parametric statistical tests. The word parametric is used in different contexts to signal different things but, essentially, it means that these tests make certain assumptions about the parameters of the population distributions from which the samples are drawn. When these assumptions are met (with varying degrees of tolerance to violation), the inferences are accurate, powerful (in the statistical sense), and are usually quick to calculate. When those parametric assumptions are violated, though, parametric tests can often lead to inaccurate results.
We've spoken about two main assumptions in this chapter: normality and homogeneity of variance. I mentioned that, even though you can test for homogeneity of variance with the leveneTest function from the car package, the default t.test in R removes this restriction. I also mentioned that you could use the oneway.test function in lieu of aov if you don't have to have...