The formulas in R have their own mini-language. You can have some detailed information in the R session with
which you can also find here.
For the sake of the example, let's say that you predict $Z$ from $X$ and $Y$ and let's drop the error terms.
$Z \sim X + Y$ means that you fit and additive model $Z_i = X_i + Y_i$
$Z \sim X * Y$ means that you fit a model with interactions $Z_i = X_i + Y_i + X_i Y_i$
$Z \sim X / Y$ means that you fit a model with "nested" interactions $Z_i = X_i + X_i Y_i$
$Z \sim X : Y$ means that you fit a model with only interactions $Z_i = X_i Y_i$
You also have the $-$, ^ and $\%in\%$ operators described in the help page. The $-$ operator is mostly used for removing the intercept and the others are redundant with the ones shown above.