Marginal Effects

The marginal effect of a regressor $x_j$ is the change in $\Pr(y = 1)$ from a one-unit change in $x_j$, holding all else fixed. In LPM the marginal effect is the coefficient $\hat\beta_j$. In logit and probit the marginal effect depends on where you are on the S-curve: $\partial \Pr / \partial x_j = g(\mathbf{x}'\boldsymbol\beta) \cdot \beta_j$, where $g$ is the link function's PDF (dlogis or dnorm in R).

When to use

Whenever you need a probability change from a logit/probit regression — raw coefficients only tell direction. Two conventions: the marginal effect at the mean (MEM) evaluates $g(\bar{\mathbf{x}}'\hat{\boldsymbol\beta}) \cdot \hat\beta_j$ at the sample means, and the average marginal effect (AME) averages $g(\mathbf{x}_i'\hat{\boldsymbol\beta}) \cdot \hat\beta_j$ across all observations. R's margins::margins(model) returns the AME with proper standard errors. See the Computing marginal effects (logit / probit) recipe.