What types of 'market mechanisms' are said to exist besides auctions?
Other methods or mechanisms by which goods are traded in a market economy: 

1. Seller posts price.

Probably the most familiar method to most people. Examples: McDonald's posts a price of \$1 for a hamburger, an airline posts a price of \$400 for a ticket, etc. Anyone who's willing and able to pay the posted price gets the good.

2. Buyer posts price.

Previously somewhat uncommon, but thanks to the internet most people are now familiar with this method. Example: Buyer of a ticket to tonight's game posts the price he's willing to pay on craigslist. Anyone who's willing and able to sell at the posted price gets to sell.

3. Seller and buyer both post prices.

Mr X announces he's willing to sell one Berkshire Hathaway share for \$300,000. Mr Y announces he's willing to buy one for \$300,000. The New York Stock Exchange (the intermediary) then matches them together and executes the trade.

4. Bargaining.

No prices posted. Typically, the buyer approaches the seller, the seller sizes up the buyer, offers a price, and the two then higgle and haggle till a mutually agreeable price is reached. Such bargaining could be over a US\$0.20 bottle of water on the streets of Hanoi or a billion-dollar takeover deal on Wall Street.



I'm not sure if your book's characterization of auctions is quite right. For example, a trade on the NYSE is quite anonymous but I don't think we could consider it an auction. (Of course, one can define an auction so as to subsume all of the above mechanisms, but in that case we have wasted the word auction.)