What do exchanges base their Bitcoin and alt currency prices on? What makes them say that they will sell bitcoins for certain amount? Do they increase and decrease their price based on what other exchanges are selling for?
An exchange doesn't set a price at all. It provides a framework for customers to buy and sell coins to each other, at whatever prices the buyers and sellers find mutually agreeable. The exchange itself does not buy or sell, and the prices it quotes are just a report of what its customers are doing right now.
Also, markets typically show the Lowest ask and the Highest Bid.
You can see some of the depth of the market on charts like this http://www.bitcoincharts.com/markets/virtexCAD.html (should show where volume of buys and sells are)
The difference between the Bid and the Ask is the Spread. Major financial markets have market makers. These people (mostly robots now) will buy and sell an inventory of stock and try to capture that spread to make a profit. The more liquid the market the tighter the spread gets.
As far as I know there are no officail professional Bitcoin Market makers yet.
You set the price, and other user's decide if they want to buy it at whatever price. If you saw 1 btc selling for $10, you'd buy it asap right now wouldn't you? And what if you saw it at $1 million, you wouldn't buy that as well right? Exchanges just keep track and process orders people set. If set a sell order for $1 million right now, no one would fulfill it until it was reasonable for them, maybe sometime in the future.
The answers listed previously, are accurate with exchanges. Then you have coinbase.com which is not an exchange its more of a conversion mechanism. For BTC to Fiat and vice versa. I came to this stack based on a query as to how they replenish their supplies during rallies like the current ATH we are seeing now.
I think exchanges and sites like coinbase buy and sell from each other. They may do this via a large exchange like bitfinex or stamp, or even smaller exchanges. This is also most likely automated and built upon algorithms that watch for the most favorable exchanges and maximized profits. Large exchanges have lots of coins that can be bought and sold over and over again.