Instant-Runoff Voting (IRV) and Approval Voting (AV).
Both systems seek to solve a simple problem with the system that the United States (and most nations) currently use, which is called Plurality Voting (PV). That problem is that PV forces voters into two opposed camps, each camp taking one candidate as their "champion". This results in what often turns out to be a maximum of unhappy voters. For example, if you place people's views on a scale from 1 to 10 with 1 being the most liberal and 10 being the most conservative, PV tends to foster candidates that fall around the 2-4 and 7-9 range However, if your goal is to elect someone who represents the majority of Americans, it would certainly make much more sense to elect someone that falls into the 4-7 range and let the end-points tug that center-line back and forth. The problem is that centrists are essentially blocked out of PV systems because they are seen as "spoilers" for the extremes.
In AV and IRV, this problem is solved by allowing people to vote for as many candidates as they wish. In IRV, you rank your choices. So in a race between candidates A, B and C, you might put a "1" next to B and a "2" next to C. This says two things: that you prefer B to C and that, in no circumstances, would you vote for A.
In AV, you simply select every candidate that you like (or, to look at it the opposite way, you vote for everyone except the candidates that you will not accept).
The tallying methods for the two are very different. In approval voting, the person with the most total votes wins. In IRV, you hold "runoff" votes until someone has a true majority.
To explain that last one, let's imagine that you're not using IRV, but rather a simple PV system like we use today. Everyone votes for one candidate. If no one gets above 50% of the votes, then you throw out all of the votes, disqualify whoever got the least number of votes and hold a new election with the remaining candidates. You do this until you either end in a multi-way tie between all remaining candidates, or someone has more than 50% of the vote. That's a basic runoff vote, and it's what IRV does. But the "instant" part of IRV means that, instead of holding a new election, you take everyone whose #1 candidate is disqualified because they got the least votes, and shift all of their votes up one, so they have a new #1. By doing this, you get a new result and you can repeat the process again.
There's a little bit of complexity in this sort of instant runoff, but it doesn't affect the voting. For example, what do you do when you get down to 3 candidates with the bottom two tied at 30% of the vote and the remaining candidate at 40%? One definition says that you disqualify the two at 30% and re-run the election with just one candidate. The result of that depends on how many people didn't rank every candidate. For example, if you remove the bottom two who are tied and end up with 49% of people voting for the remaining candidate and 51% with no vote, then the vote could be considered "failed" because no one got a true majority. On the other hand, you can disqualify all voters who have no remaining votes and then the remaining candidate would be considered to have 100% of the votes.
Either way, you figure these things out ahead of time and therefore just have to apply the rules to the votes you get. No one has to worry about what happens in such a case as long as they do vote.
In terms of outcome, each system has some pros and cons. AV gives you the mathematically best result in the sense that the distance between a person's desire for a candidate and the winner will be lowest on average with AV. It will be slightly greater with IRV and largest with PV (which we use now), so both systems are an improvement over what we have, but AV is the largest improvement.
AV has one large problem, however. It's not only possible, but fairly common that the winner of the election will not be the person who is most people's first choice. Since this is also the strength of the system, it's considered acceptable by most people who study voting systems. However, when you have a Gallup Poll that shows that 60% of the population wants candidate A to win and candidate B wins, it's hard to explain that that really was the will of the people...
IRV doesn't have this problem, since any candidate who has a simple majority of #1 votes going in, just wins in the first round. It's only when no one has 50% of the vote going in that they can lose, even though they are the "favorite" according to a PV, but then you go back to the confusion over why.
Both IRV and AV are more resistant than PV to false voting. That is, if you vote because you want to "spoil" someone's chances, rather than just favor your own choice, then PV is where you will have the most success. In IRV it's harder to spoil the result, but there are strategies for false voting that have some impact. In AV, you have essentially no way to skew the vote other than to support the candidates that you want to see win.
There are also other ways to tally both IRV and AV type systems, but that's beyond the scope of this article.
So, what's the best system?
It really depends on what your goals are. If you want to polarize the field down to a small number of choices that sort of "summarize" the various positions, IRV works well. This is handy in pre-elections like party primaries. If you just want to choose the candidate that will best represent all of the voters, then AV is the better system (as long as you can tolerate the consequences of a popular candidate losing the vote).
Either way, the U.S.'s current election voting system is a terrible way to run anything more complex than audience participation in a comedy routine.