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Sure. Let's start with dealers. A dealer is a party that stands ready to buy or sell something (in this case bitcoins). When you want to transact, you ring up the dealer and ask for a quote. The quote is composed of a price at which the dealer will buy bitcoins and a price at which the dealer will sell bitcoins. If you like the price, you do the deal. So throughout the day, the dealer is constantly buying and selling. The difference between the buy and sell price gives the dealer his profit. If a dealer decides he is building up too much inventory, he may, for example, lower his bid price (widening out the spread) so that fewer parties will be interested in selling to him. That's the basic idea of a dealer.

Now exchanges. The idea with an exchange is to match up a plurality of buyers and sellers. Exchange participants can send in orders to buy or sell a specific quantity at a given price (these are called limit orders). These orders are all combined and sorted by price into a limit order book. If you look around at some of the bitcoin exchange sites most of them will show you the state of the book†. When you look at a quote at an exchange you are really looking at the highest priced buy order (or the aggregate of all orders at that price) and the lowest price sell order. When market participants wish to transact immediately, they can send in a marketable order, meaning that buyers will pay the best offered price or sellers will pay the best bid price. Marketable orders "cross the spread". The exchange is responsible for updating the book and distributing it to market participants, but the exchange itself doesn't do any buying or selling. Exchanges take a percentage fee of each transaction which is the source of an exchange's profits.

Let's say I'm a bitcoin miner. My mining machinery creates bitcoins out of electricity but I still need dollars to pay the utility bill. This makes me a natural seller of bitcoins. When I go to sell my bitcoins, I want to get the best price for them. I might post a limit order to sell at what I think is a reasonable price on an exchange and wait for my order to get hit. I might also look at the lowest offer price and decide to post my bitcoins for sale at a penny less in order to increase the probability of getting my order filled quickly. This kind of limit order trading is only available at an exchange. If I could only transact with a dealer, I'd have to either cross the spread and accept the dealer's bid or not do a deal at all.

† For example: https://www.bitstamp.net/market/order_book/