Hi Erich,
You run an independent samples t test when you want to compare the means between two groups. As an example, if you want to compare males to females on some outcome variable, you would run an independent samples t test.
If you want to compare the means on a variable that contains more than two groups, however, you would run a one-way ANOVA. As an example, if you want to compare the means of individuals from different classes (e.g., working class, middle class, and upper class), you would run an ANOVA.
Factorial ANOVA also allows you to look at the effects of multiple grouping variables (called factors) at the same time. Say you want to see the effect of both gender and class on some outcome variable, you would run a factorial ANOVA (in this case, a two-way ANOVA because you have two grouping variables). An independent samples t test only allows you to look at the effect of one grouping variable (with only two levels) at a time.
Hope this helps!
-Mike
