How do prodrugs exactly work? Are there multiple varieties of prodrugs?
Prodrugs are extremely common; we often don't even think of some of them as prodrugs unless we're also familiar with the metabolite. Psilocybin, for example, is a prodrug; the active molecule is psilocin. Aspirin is a prodrug of salicylic acid (though there are also other active metabolites).
Prodrugs can be categorized according to where they are converted to active forms: intracellularly in target tissue or other organs like the liver, or extracellularly like in the gut or in the bloodstream. The Wikipedia link at the beginning of this answer includes a table with some examples.
Prodrugs may be converted to active forms by enzymes or by environmental conditions in the body, like pH. I'd argue they are diverse enough that it doesn't make all that much sense to study them as a category, rather that in any pharmacological class of drugs you should be aware that they may be administered in a modified form, often to improve bioavailability or storage stability.