With one-way ANOVA, we only had one thing of interest - the group that the participants belonged to. This thing, which we could call group, is known as a FACTOR. Moreover, the factor Group had three LEVELS, i.e., there were three groups.

ANOVA can also be used when there is more than one factor, each of which has two or more levels. For example, if we had a group of men and a groups of women, we would have a factor called Gender with two levels (male and female). Moreover, if half the people had blond(e) hair and half brown, we would have another factor called Hair Colour which again has two levels (blonde and brown). If these people all take a test, their groups will give us four means, one for each of the following groups: blond men, brown-haired men, blonde women, brown-haired women. A two-way ANOVA (because there are two factors, Gender and Hair Colour) will tell us whether these means differ significantly.

Again, the analysis produces F-ratios, which represent the amount of variance accounted for by the factors relative to the amount of random error variance. The difference is that now we have an F-ratio for each MAIN EFFECT as well as one for the INTERACTION.
In this analysis, we would have an F-ratio for the main effect of Gender and an F-ratio for the main effect of Hair Colour. If the first is significant, then men are performing differently from women. If the second is significant, then blond(e) people are performing differently from brown-haired people. It is possible for none, one, or both main effects in this analysis to be significant. For example, you might just get one significant main effect of Gender, because men score lower than women.

The interaction anaylsis also produces an F-ratio. This is a slightly tricky concept. A significant interaction occurs when the difference in one of the factors is affected by the other factor. For example, imagine we got these data:

blond men score 50 brown-haired men score 80 blonde women score 50 brown-haired women score 20

Here there is an interaction - the effect of gender isn't straightforward as it is different for each hair-colour. ANother equally valid way of saying this is that the effect of hair colour isn't straightforward as it is different for each gender. This is an interaction.

ANOVA is done with Analyse > General linear model > Univariate

The dependent variable goes where you'd expect it to in the dialogue box. The variables that code groups (e.g., 'gender', 'class', 'drug dose', etc.) go in the "fixed factors" box. The other buttons on the dialogue box allow you to get things like descriptive statistics, graphs, and post-hoc tests. Remember, for post-hoc tests you usually use the Tukey HSD test.