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The bids and offers made in the market set the price, just like all currencies. Right now the market is too small to be stable like most currencies (not enough inertia, too many events if large relative significance, not enough arbitrageurs), but the wild swings you see are not due to the free will whims of the exchange managers, they're due to the market's youth and size.

"That makes me feel uneasy about the conversion rate of bitcoins to dollars. What is that really based on?"

I suspect that it would be based on the number of people trying to use Bitcoin as a payments network at any given time.

If their price was wildly out of with the rest of the market, they'd either run out of bitcoins or out of USD pretty fast.