IIUC, when someone “makes a market” they make a credible offer to buy or sell something, presumably more than once.

So in The Big Short, when Jared says to Vinny, “you want me to make you a market?” what he’s asking is whether Vinny wants to buy what Jared is trying to sell for the first time (credit default swaps).

In an exchange context, when you place a limit order to buy or sell, you’re called the market “maker”. Your counterpart, who fills the order, is the market “taker”.