Who are the market makers in options trading?

Market makers tend to be individuals that work for financial institutions such as banks and brokerage firms. They are contracted with an exchange for designated securities (a Designated Primary Market Maker or DPM). Per requirements, he sets his Bid/Ask spread which has maximum limits and he must transact a set number of contracts with anyone willing to transact at his posted prices.

A common misunderstanding is that a market maker sets the bid/ask price and that you must trade at that price. Any market participant can offer a higher bid or a lower ask price than the DPM and become the market on one side of the NBBO quote (you can be on both sides with equities but not with options). If you do so, you are acting "like" the market maker since yours is the best bid or the best ask. Temporarily, you are the market maker (on that side) until you get a fill or someone offers an even better price. If a counterparty is willing to transact at your price, your trade is executed and the DPM has nothing to do with the trade.

Though it's far from complete, here's the CBOE Symbol Directory for equities which lists the name of the DPM.

Citadel and Susquehanna are two major market makers. Both used to be statarb funds which expanded into MM over the last 10 or so years.