I'm not sure the above writeup gives the full story, so allow me to jump in.
As stated above, Arrow puts forth 4 burdens on an ideal voting system for a democracy:

  1. Universal Domain - The system should be able to contain any possible social choice
  2. Independence of Irrelevent Alternatives - An individual preference stated in binary terms (one over the other) shouldn't affect his/her or other's ordering of other preferences. Ex: Liberal voters in America considering the choice between Ralph Nader and Al Gore shouldn't have to worry about inadvertently electing George W. Bush with their vote for Nader.
  3. Pareto Optimality - If every individual votes the same way, the society is decisive.
  4. Nondictatorship - More than one person's vote should be used in determining social choice

The Arrow Theorem, in simple terms, says that no voting system can satisfy all 4 above criteria. The only way voting systems work, then, is by relaxing one of the criteria (which in practice is usually universal domain). The proof of the theorem is rather simple and elegant. It makes use of two principles:

The problem? If each larger group is subdivided into smaller groups of identical preference, the second principle says that group can be subdivided as well. Thus, when the process is over we arrive at the vote of one person. This person is the dictator of the system, thus failing the fourth criterion.
Sen, Amartya. Rationality and Social Choice. American Economic Review, March 1995. Vol 85, No. 1