A pyramid scheme is a recruitment-driven arrangement in which participants pay money in and are rewarded chiefly for bringing in more paying participants. The most widely cited description in United States law comes from the Federal Trade Commission's 1975 Koscot decision, which characterized such enterprises by "the payment by participants of money to the company in return for which they receive (1) the right to sell a product and (2) the right to receive in return for recruiting other participants into the program rewards which are unrelated to the sale of the product to ultimate users." The second half of that sentence is the whole test. A right to sell something is not the problem; a reward that arrives for recruiting rather than for selling is.
The Commission concluded in the same decision that recruitment with rewards unrelated to product sales was "nothing more than an elaborate chain letter device", worthy of "categorical condemnation" under Section 5 of the FTC Act for its "inevitably deceptive representation … that any individual can recoup his or her investment by means of inducing others to invest." That framing matters because it explains why no product is required: a chain letter has none, and the structure is identical.
A pyramid scheme is not the same thing as a Ponzi scheme, and it is not the same thing as multi-level marketing. In a Ponzi scheme investors believe they have bought a strategy and generally have no idea that the money paid to them came from later investors. Multi-level marketing is a lawful and diverse way of distributing products through a network of sellers, and whether a particular company's compensation structure has crossed into a pyramid is a separate, fact-intensive question.