Both mutual funds and hedge funds are pooled investment vehicles. This means that investors money is added to the "pot" and invested as a single portfolio. The differences in mutual funds and hedge funds primarily have to do with regulation and investment strategies.
Mutual funds are regulated under the Investment Company Act of 1940, overseen by the SEC. Hedge funds are private partnerships where each investor is a limited partner. The portfolio manager is typically the general partner. Therefore, hedge funds are not regulated under the ICA of 1940.
The other major difference in mutual funds and hedge funds is investment strategy. Most mutual funds pursue a relative return strategy. This means the mutual fund measures its performance versus an index or sector of the economy.
Hedge funds persue an absolute return strategy. This means generating positive returns anually regardless of market direction or economic situations. This can be done via literally hundreds of individual strategies.