The price of Bitcoin is determined by supply and demand. On a Bitcoin exchange, people who want to buy Bitcoin and people who want to sell Bitcoin place orders at their desired price, and a sale is completed when a match is made.
Unlike dollars or yen, Bitcoin is not backed by a government, central bank, or physical asset; Bitcoin has value because people believe it has value.
Bitcoin is based on the idea that it should not be allowed to have inflation, so the amount of Bitcoin that will be issued is pre-determined. If there is demand in excess of the available amount, the price will go up. On the other hand, if there is less demand, the price will fall.