My understanding is, there are different tests to run such as ANOVA, Pearson's Correlation, Chi-Square. Choosing these tests is dependent on if the features / responses are categorial / continuous. And then each test has it's own test specific ways to measure "importance", for example: https://chrisalbon.com/machine_learning/feature_selection/anova_f-value_for_feature_selection/
The MAIN test that I've seen though is Random Forest. I've used Random Forest's to get feature importance graphs like this: https://imgur.com/a/FY3muq1.
However, I'm not really sure what is like the high-level or industry standard way to respond when someone asks something like "how would you determine the features that strongly affect whether or not someone buys a product?". In the past I've defaulted to discussing the RF feature importance projects I've done, but
1) I'm not super confident regarding the math behind RF feature importance
2) defaulting to RF seems super specific and might make it seem like I don't understand feature importance in general
3) I don't know what to show as a final result other than just the feature importance graph. Like for example, I don't know how to get really quantitative and make a claim like "Feature A is 2x more influential / important than Feature B".