How is the price of bitcoins determined?

Two words: supply and demand. The price rises when the demand for bitcoins grows and falls when the demand decreases.

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So Bitcoin's price shows how many people are willing to pay for it at a given time based on their expectations regarding its future value. It is the same as with the price of another currency, or fiat money or cryptocurrency.

Since Bitcoin is still young and its market capitalization is minuscule compared to that of the US dollar or the euro, its price is still very volatile. It is expected to become less volatile when the market is maturing. The nice thing about it is that nobody knows when that will happen.

Why the price can rise:

The offer is limited - it is capped at 21 million. There are currently more than 16.7 million bitcoins on the market. Achieving the limit of 21 million can take more than 120 years.
Technological development: improvements in security, anonymity, usability and user experience plus new functions can increase the use of Bitcoin and positively influence the price.
Favorable regulations: if governments support the use of Bitcoin, that would increase its price.
New business solutions can arise that increase the demand for Bitcoin.
Public perception: if more people see Bitcoin as an attractive form of money or assets and confidence, that would increase demand and thus also the price.
Why the price can fall:

We can see the price of bitcoin drop in response to technological problems, ecosystem errors, unfavorable regulation, restrictions, negative media attention or other reasons why people lose confidence in the currency.

Also, remember that Bitcoin is traded in pairs like any other currency. So trends in the price of bitcoin may differ slightly when it is traded for US dollars than, for any other currency.