I don't understand why the conversion rate for bitcoin changes wildly. Why does it?
As with any traded commodity the price largely depends on the confidence of the buyers and sellers in the future evolution of the price. Basically you pay whatever you feel it is worth. Like stocks this changes over time, as the company or the bitcoin economy in our case evolves.
The wild price swings are not that really that surprising then when it is mostly sentiment driven. Any sudden move in either direction may cause uncertainty, triggering reactive selling or buying that will again trigger more reactions. This cycle is then broken by people seizing the moment to buy in cheap or sell high, stopping the momentum, possibly triggering a move into the opposite direction.
Because the demand for bitcoins is largely driven by speculation, i.e. people buy bitcoins in the hope that the conversion rate will rise so that they can make a profit.
Such speculative investors tend to exhibit herd behaviour: the more the rate falls, the more of them will sell, to realize gains or cut losses. And the more it rises, the more people want to get in on the easy money. Thus, a large percentage of speculative investors always leads to a high volatility.
Conversely, People who buy bitcoins because they want to use them as payment for something else are far more likely to hold on to their bitcoins when the exchange rate falls, or even buy more.
Bascially, any regular business that takes bitcoins as payment represents a non-speculative demand and a stabilizing force on the exchange rate. This adoption is what will decide the long-term viability of bitcoin as a currency; a spectacularly rising exchange rate is actually harmful in that regard because it attracts mainly speculative investors.
Like all markets the price at each individual exchange is determined by supply and demand. However in most markets when price discrepancies appear between them, it means that there are arbitrage opportunities, and the gap will quickly be closed because people can turn a profit by buying on the exchange trading lower and selling on the exchange trading higher.
This is not the case with bitcoin because of the difficulty in moving fiat money between the exchanges. For instance when bitinstant happens to be working, you can move USD from MtGox to Btc-e at a 1.49% fee. Because the fee is so high, the percent difference between the two exchange will generally be in the range of [-1.49, 1.49%]. For example say Btc-e is trading at 150 USD and bitinstant is working. Then you can expect the price of BTC on MtGox to be anywhere between $147.76 and $152.23, because if it where anything outside that range people could turn a profit and it would quickly put the price back in the range.
In times like these (7/8/13) when both bitinstant is down and USD withdrawals are not being allowed from MtGox, you can see crazy large discrepancies in the area of 7%.