# Coefficient marginal to interactions in linear regression

Consider this model:

$$y = \beta_0 +\beta_1x_1 + \beta_2x_2 + \beta_3x_1x_2 + \varepsilon$$

Somebody told me today that the coefficient for the main effect of $x_1$ (i.e., $\beta_1$) will be 'marginal to the interaction'.

Could somebody explain what 'marginal to the interaction' means?

I wouldn't phrase it that way. The common meaning of "marginal" in statistics (and the way I use it) is roughly 'ignoring'. In other words, the marginal effect of $x_1$ is the effect of a 1-unit change in $x_1$ on $y$ ignoring all other possible variables. That is not what $\beta_1$ represents in that model. (Note that they may be using the terminology differently than I do, however; that happens.) To understand the idea of marginal (in the sense of ignoring) in regression better, it may help to read my answer here.
Instead, I would say the main effect of $x_1$ (i.e., $\beta_1$) in your model is the effect of a 1-unit change in $x_1$ on $y$ when $x_2=0$. To get a clearer sense of this, it may help to read my answer here (that answer in particular explains in detail what $\beta_1$ means when an interaction is present).