Bitcoins do not have value as a physical commodity like gold and are not widely accepted as legal tender like dollars. Rather, Bitcoin appears to have value for the following reasons:
It is popular. In short, people accept and trade in Bitcoin because other people accept and trade in Bitcoin. It is recognized and accepted as a currency by many.
Bitcoin is decentralized and limited. This is a major factor for many Bitcoin users. Bitcoin is hard for governments to trace and tax. Also, unlike fiat money produced by central banks, there is a cap set on total Bitcoins, limiting how much the currency can devalue through inflation.
Bitcoin acts like an equity investment. The market value of Bitcoins has had wild swings in value and even a market cap.
Bitcoin is a social network. The Bitcoin "community" is active and acts like other online social networks.