Affinity fraud is an investment fraud that reaches its victims through a shared identity rather than through a cold approach. The Securities and Exchange Commission defines it as investment scams "that prey upon members of identifiable groups, such as religious or ethnic communities, the elderly, or professional groups", and adds that the promoters "frequently are - or pretend to be - members of the group."
The word describes a method of distribution, not a distinct financial mechanism, and that is the single most useful thing to know about it. The SEC's own entry says plainly that many affinity scams involve a Ponzi scheme or a pyramid scheme, in which money from new investors is used to pay earlier ones. So the question "is this an affinity fraud or a Ponzi scheme?" is usually answered "both": the Ponzi arithmetic explains where the money goes, and the affinity explains how the scheme found people willing to hand it over.
A note on the name, because the SEC now uses two. Its glossary still carries an entry headed "Affinity Fraud", which is where the definition above comes from. Its consumer page on the same conduct has been retitled "Investment Scams Targeting Groups" and no longer uses the older phrase. "Affinity fraud" remains the term of art and the term people search for; the plainer description says the same thing.