Two-way Analysis of Variance
Danger - Danger - Warning!! Don't Say All the Means are Different Because F-ratios are Significant.
WRONG!
You will need to do comparisons. Check the One-way Anova workshop for the rationale of doing this. It is very similar for Two-way Anovas.
Variances can be Turned into Proportions
(Effect Size)
Here's the deal - if you know the total variance - you can determine what percent of the total variance is related to the Mean squares for each term in your analysis (A, B, AB) variance as compared to the Within Groups Variance.
Many folks like a coefficient called Omega2 the best.
Bottom Line
If you have a two independent variables, your goal is to see if each variable has an effect and whether they interact to give different effects for one variable dependent on the level of the other.
Check the FAB ratio first.
Look at the Graphs!
Do comparisons to find out what groups are significantly different.
Calculate an effect size measure like Omega Square to see how much variance A, B and AB explain.
A parting nuance
One can have 2-way, 3-way, 4-way and N-way designs. The idea is all the same. In a 2-way design (AB) - you worried about if the effect of A depended on B. In a 3-way design you have ABC and you can worry about whether the AB interaction depends on the level of C (or AC on B or BC on A). What this means is that for a level of C, the AB graphs are different. Here are some pictures.
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