Partitioning Information
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 wondered if the effect of A depended on B. In a 3-way design, you have ABC and you can ask 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.