Does a market maker sell (buy) at a bid or ask price?

I think your confusion has arisen because in every transaction there is a buyer and a seller, so the market maker buys you're selling, and when you're buying the market maker is selling.

Meaning they do in fact buy at the ask price and sell at the bid price (as the quote said).

The answer posted by Kirill Fuchs is incorrect according to my series 65 text book and practice question answers. The everyday investor buys at the ask and sells at the bid, but the market maker does the opposite. THE MARKET MAKER "BUYS AT THE BID AND SELLS AT THE ASK", he makes a profit from the spread. I have posted a quiz question and the answer created by the Financial Industry Regulatory Authority (FINRA).

To fill a customer buy order for 800 WXYZ shares, your firm requests a quote from a market maker. The response is "bid 15, ask 15.25." If the order is placed, the market maker must sell: A) 800 shares at $15.25 per share. B) 800 shares at $15 per share. C) 100 shares at $15.25 per share. D) 800 shares at no more than $15 per share.

Your answer, B) sell 800 shares at $15.25 per share, was correct!

A market maker is responsible for honoring a firm quote. If no size is requested by the inquiring trader, a quote is firm for 100 shares. In this example, the trader requested an 800-share quote, so the market maker is responsible for selling 8 round lots of 100 shares at the ask price of $15.25 per share.

Market Makers are essentially just there to process the buys and sells of traders, so just like you and I buy and sell at the ask and bid prices they do to. They are just completing the process of making our orders a reality. Market makers are just representative of brokers, meaning that when you place your order at ask or bid, you are placing that particular brokers order at ask or bid. People often say that certain brokers have too many shares and claim that they are games when really that just means that there happen to be a lot of people using a particular broker all at once, or more troubling, perhaps even company execs using a broker, to sell a large amount of shares.

The everyday investor buys at the ask and sells at the bid but the market maker does the opposite

This quote from Ryan's answer is misleading; it has nothing to do with being either an investor or a market maker. It is dependent on the type of order that is submitted. When a market trades at the ask, this means that a buy market order has interacted with a sell limit order at the limit price. When a market trades at the bid, this means that a sell market order has interacted with a buy limit order at the limit price.

An ordinary investor can do exactly the same as a market maker and submit limit orders. Furthermore, they can sit on both sides of the bid and ask exactly as a market maker does. In the days before high frequency trading this was quite common (an example being Daytek, whose traders were notorious for stepping in front of the designated market maker's bid/ask on the Island ECN).

An order executes ONLY when both bid and ask meet. (bid = ask)

This quote from Kirill Fuchs is completely incorrect. A transaction occurs when an active (marketable) order is matched with a passive (limit book) order. If the passive order is a sell limit then the trade has occurred at the ask, and if it is a buy limit the trade has occurred at the bid. The active orders are not bids and asks.

The only exception to this would be if the bid and ask have become crossed.

When a seller steps in, he does so with an ask that's lower than the stock's current ask

Almost correct; he does so with an order that's lower than the stock's current ask. If it's a marketable order it will fill the front queued best bid, and if it's a limit order his becomes the new ask price. A trade does not need to occur at this price for it to become the ask.

This is wrong, market makers are the opposite party to you so the prices are the other way around for them.

This is wrong. There is no distinction between the market maker and yourself or any other member of the public (beside the fact that designated market makers on some exchanges are obliged to post both a bid and ask at all times). You can open an account with any broker and do exactly the same as a market maker does (although with nothing like the speed that a high frequency market-making firm can, hence likely making you uncompetitive in this arena). The prices a market maker sees and the types of orders that they are able to use to realize them are exactly the same as for any other trader.