Approval voting has several compelling advantages over other voting procedures:
Citizens for Approval Voting is an educational and outreach organization formed in 2003 whose attractive web site provides good information about and arguments for Approval voting.
There is also a sister political organization Americans for Approval Voting which is dedicated to getting Approval Voting adopted for single officeholder elections in America. I think it is clear that this would put our democracy on much better footings and I heartily endorse Americans for Approval Voting.
Read on for more background on voting and evidence for all these claims.
See also Approval Voting, J. Econ Perspectives 9(1), Winter 1995, by Robert J. Weber, who came up with the name "Approval Voting" in 1971 and was one of its discoverers.
Other good references:
When there is just one winner in an election, a wide array of alternative voting reforms have been put forth. Lest you get lost in the details, keep in mind that nearly everyone agrees that the traditional single-vote-plurality system does the worst job of picking the candidate that the voters prefer.
The first solidly researched comparison that I saw, which includes computer simulations based on models of voting behavior, was Making Multicandidate Elections More Democratic, by Samuel Merrill, Princeton University Press, 1988. It concludes that the most reliable systems for meeting the "Condorcet" and "maximum social utility" criteria are approval voting and the Instant Runoff Vote (also known as IRV or Preferential Voting) and traditional runoff methods. Among these three, approval voting ranks slightly higher and is much easier to implement. IRV suffers from significant problems in real life. This was demonstrated in the 2009 Burlington mayoral contest, where a candidate that clearly beat each other candidate head-to-head ended up being eliminated early on!
A more recent approach, Range Voting shares many of the advantages of Approval Voting and allows more nuanced votes, but makes ballot design and public education somewhat more difficult.
Other methods, such as the Borda Count and those based on the Condorcet tally itself, tend in practice to be vulnerable to "strategic voting". I.e., voters have incentive to vote insincerely (e.g. punish their favorite's chief rival with a last-place vote), and the resulting outcome matches the Condorcet criteria less than methods which don't reward voters for insincere votes.
And the Borda Count is supported in a book that compares it with Approval Voting: Chaotic Elections! A Mathematician Looks at Voting by Donald G. Saari
It seems clear to me that approval voting is the most practical and best system for single-winner elections. Here are some more references to other perspectives:
When there is more than one winner the situation is very different. Most countries have a different perspective on what it actually means to select a legislative body that represents the people, focusing more on representing viewpoints than geographical areas. See the Center for Voting and Democracy to learn about Proportional Representation.
No matter what voting system is used, it is important for you to learn about the candidates and issues (yahoo | NPR) and vote!