For example, a classifier is a generative model if it models p(example, label) -- which is sufficient to also calculate p(label | example) if you want -- rather than just modeling p(label | example) alone.
Similar example in translation: a generative translation model would model p(french sentence, english sentence) -- implicitly including a language model of p(french) and p(english) in addition to allowing translation p(english | french) and p(french | english). A non-generative translation model would, for instance, only model p(french | english).
I don't exactly understand what this judge meant by "generative", it's presumably not the technical term.