Indeed. say that one factor is gender being female (D1) and the other is having a University degree (D2) and your response variable is wage (y) presumably in logs. Then you could of course estimate the model y=B1*D1+B2*D2+e (where e is an error term) and get the main effects that is the gender and the University effect on wages, but you may equally well estmate y=B1*D1+B2*D2+B3*D1*D2+e where B3 would estimate the additional effect of University degree for females on wages. This is readily done in a regression framework although in this simple example it is sufficient to compute the mean wages in the four groups in your sample, namely males/females and w/wo University degree.

Yes it is perfectly possible. You need to include the interaction term into your model. The type of the model will depend on the type of dependent variable and your hypothesis.

Indeed. say that one factor is gender being female (D1) and the other is having a University degree (D2) and your response variable is wage (y) presumably in logs. Then you could of course estimate the model y=B1*D1+B2*D2+e (where e is an error term) and get the main effects that is the gender and the University effect on wages, but you may equally well estmate y=B1*D1+B2*D2+B3*D1*D2+e where B3 would estimate the additional effect of University degree for females on wages. This is readily done in a regression framework although in this simple example it is sufficient to compute the mean wages in the four groups in your sample, namely males/females and w/wo University degree.

Yes - it is common to run this kind of model as an analysis of variance (ANOVA), though this is just a special case of multiple regression (usually with a slightly different parameterisation). Depending on software running it as an ANOVA may offer a wider range of options.

yes, you can use regression with dummy variables to get your models.

You can use regression model or anova model depends on your experimental design.

How do you interpret it?

Bambor et al. (2006) says that you should interpret the conditional effect, or the impact of D1/y=beta1+b3(meanD2). And in this case D2 is categorical = 1 or 0 so the impact of D1 on y when D2=1 is beta1+beta3.

Kenneth Carling What if your D2 is Education, instead of university education, whose value is 0 for primary education, 1 for high school, 2 for university level. let's have 0(primary education) as benchmark. What your model should look like? Thank you Sir.

Virak Khiev Well your question is somewhat off-topic as the original question pertained to dummy variables, i.e. factors with only two levels. Your question refers to three levels. However, the basic intrepreation is the same, although now it refers to multiple subgroups.

Kenneth Carling I'm sorry if my question is irrelevant. let assume that I'm proposing a new topic which is an interaction between a dummy variable and a polytomous variable.

I'm curious about how your model (y=B1*D1+B2*D2+B3*D1*D2+e ) will change when incorporate a higher level variable as given above. Would you please demonstrate? I'm fine with interpretation, I guess. Thank you!

Virak Khiev The model would look exactly the same. How it should be specified in a specific statistical software may however vary. In for instance Minitab and R, your case and Mia Andrea Ortiz Soriano's case would be identically specified. In other software they may look different in terms of specification.

Kenneth Carling so in that case, you don't need to convert the polytomous variable into dichotomous variable with a 0/1coding?

Virak Khiev Again, it depends on the software. R and Minitab would not require it. Also for SAS I tend to recall it not being needed. But computationally the underlying structure implies representing a factor with k levels by k dichotomous variables. I suggest you refer to the manual for the software of your preference as to judge what is required.