Why is the electoral college needed?

There is significant debate in regard to whether the college is actually necessary, and in fact there has been since its inception. However, because you simply asked why it is needed, I'll stick to that side of the debate in my answer.
The electoral college was created as part of a compromise with some of the less populous states to avoid something known as the "Tyranny of the Majority," where the handful of states that had the majority of the population would completely dominate national politics. It prevents this by tying the number of electors in the electoral college to the number of representatives and senators that the state has in Congress. Because each state is guaranteed to have at least three representatives in Congress, (Two Senators and One Member of the House) they are also guaranteed at least three electors as well. This saves states like Wyoming and Vermont from having no power whatsoever in the presidential election.
This does result in an unequal degree of representation. A Wyomingites vote, for example, is worth roughly 14 times that of a Californian's. However, many see that as a necessary evil to prevent those states with the most people from completely deciding the elections, and potentially ignoring the needs and wishes of the rest of the nation.
There used to be additional reasons for the use of the Electoral College, such as to allow state legislatures to determine how the state would vote without needing to resort to a popular vote, but due to changes in the law, culture, and technology these are now insignificant.

Many people would argue that the electoral college is not needed and that it is more a hindrance to democracy than a help. However, separation of powers was deemed the surest guarantee of liberty, and the Constitution builds in several inefficiencies to preserve liberty for all.
In order to persuade the original colonies to ratify the constitution, a variety of incentives were thrown into the mix. Misgivings of direct election of a president caused early colonialists to hesitate regarding the type of democracy the new country should adopt. The electoral college offers a halfway point between direct democracy in which the total of all votes cast would determine the president and one in which governors or members of Congress would vote for the president. The electoral college allows a mathematical apportionment so that citizens' votes are somewhat valuable in selecting the president.
At the Philadelphia Convention, Washington, Hamilton and Madison sought a more republican democracy. With voices on either side, the goal was to prevent the executive branch from becoming tyrannical, while also ensuring the populace of individual states did not become overly empowered. The electoral college allows small, underpopulated states to have at least three votes, no matter how small the population. This is seen as a check against larger states and ensured the Constitution would be ratified.
While often seemingly flawed in its execution, the theory behind the electoral college is that this body and the means by which electors are apportioned would prevent any area of the democratic process from becoming too powerful.