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There are at least two classes of situations in which representatives are elected by election; single seat and multiple seat. In the single seat case, there is only a single office to be filled (for instance, the election of the U.S. president); in the multiple seat case, there is a number of similar offices to be filled (for instance, the election of a legislature).
Our current system is called plurality (sometimes also called first-past-the-post). The plurality system is: each voter picks a candidate. The number of ballots for each candidates is counted. Whoever gets the most ballots wins.
There are at least two big problems with plurality.
The first problem is the possibility of splitting the vote or spoilers. If only one candidate appears to be competitive for faction A, and there are multiple candidates for faction B, then faction B's votes are "split" among the different faction B candidates. Faction A has an unfair advantage because faction A's candidate will get as many votes as there are faction A voters, whereas none of faction B's candidates will get as many votes as there are faction B voters. For example, if faction A has 40% of the voters, and faction B has 60%, but there is only one candidate for faction A and there are two for faction B, both equally popular, then each faction B candidate will only get 30% of the votes, and faction A's candidate will win with 40% of the votes -- even though faction B has 60% of the votes. When one of the multiple candidates is less popular than the other, the less-popular candidate is said to be a "spoiler"; a real-life example may the 2000 presidential elections in which it is thought that Nader and Gore may have split the left vote, giving the victory to Bush.
Vote splitting is clearly a problem in the short term because it leads to a less popular faction winning elections. In the long term, it is an even bigger problem because it prevents anyone but the top two front-runners from gaining a large number of votes, because voters don't want to "waste their vote" on a spoiler. This greatly contributes to two-party dominion, because it prevents third parties from gaining an appreciable number of votes (see also Duverger's law).
The second problem with plurality is that "compromise candidates" cannot win. An example of a "compromise candidates" is one who is no one's favorite, but everyone's second choice.
For example, imagine that every voter is in one of two factions, call them Democrat and Republican, that each faction controls about 50% of the voters, and that there are three candidates for President; Bush, a Republican, Gore, a Democrat, and Mr. Second Choice, who is a centrist. Republicans like Bush, don't mind Mr. Second Choice, and hate Gore. Democrats like Gore, don't mind Mr. Second Choice, and hate Bush. With plurality, all the Republicans will vote for Bush, and all the Democrats will vote for Gore, and no one will vote for Mr. Second Choice. Which is a shame, because if Mr. Second Choice were to be elected, no one would hate him, whereas what will happen instead is that either Bush or Gore will win, and either way about half the country will hate the president.
As with vote splitting, in addition to the short-term effects, this leads to a long-term problem. The long term effect is that politicians have no incentive to appeal to voters in the "enemy" faction. You have to either be the most republican-y Republican or the most democratic-y Democrat; it does no good to be the second best Republican, even if the Democratic voters like you pretty good at the same time. This means there are incentives to demonize the "enemy" party, and no incentives not to. When one party is in power, there are no incentives not to do things that the other party hates. Therefore, plurality makes politics more divisive, and less cooperative.
There are many alternative vote counting methods that don't have the problems of vote splitting or elimation of compromise candidates. My two favorites are range voting and Condorcet (actually these are each a small family of methods). I won't go into them much here, but you can read about them at http://rangevoting.org/ and http://en.wikipedia.org/wiki/Condorcet_method.
Currently, I recommend range voting. A simple form of range voting is this: for each candidate, each voter gives them a score of either 0, 1/2, or 1. These scores are summed up, and the candidate with the highest score wins.
Our current system for electing congresspeople combines plurality with districts. Each congressional seat is assigned to a district, and a single-seat plurality election is held in order to deterimine who wins that seat. In addition to the problems with plurality, this has the additional problem that it does not provide "proportional representation". What that means is that the factional proportions in congress might not match the proportions in the electorate. For example, imagine if 60% of the voters belonged to faction A, and 40% belonged to faction B, and faction loyalties were evenly distributed geographically. In this case, using our current system, faction A would win every seat, and congress would be made up of 100% faction A. A better outcome would be if congress would be 60% faction A and 40% faction B.
An example from real life is the green and libertarian parties, which get a few percentage points of the votes but don't have any representatives in congress (because there is no single district in which they have a majority). Most likely, many more people would vote for these parties if doing so was not "wasting your vote"; my guess is that on the order of 20% of congress would be green and libertarian if we used a vote counting system with proportional representation for congress.
There are many alternative vote counting methods that provide proportional representation. My two favorites are reweighted range voting and Loring. Currently, I recommend reweighted range voting.
In both of these methods, either single-seat congressional districts are eliminated and replaced by larger districts with multiple seats per district, or congress is augmented by the addition of at-large members representing the whole nation, or both.
