One can imagine several main options:
- Single round plurality voting.
In this system the candidate with the most votes in the first round wins. This means that if there are more than two candidates running, a third party candidate can "spoil" the vote for the viable candidate most similar in ideology to him by splitting the vote of people who share their ideology, making the winner the candidate whose ideology actually has less support in the electorate than an alternative.
The virtue of this system is that it is very simple.
- Majority voting.
In this system, if one candidate gets a majority of the vote in the first round, he wins, but if not, there is a runoff of the top two candidates.
This first prevents the spoiler effect, and second, allows members of the public to have time to give more attention to the two candidates who could win, particularly if they voted for a third party candidate the first time around. This reflects the fact that people aren't always good at making hypothetical decisions and don't invest as much effort into considering second and subsequent choices as their first choice.
The downside is having a longer campaign seasons and paying for two sets of balloting and inconveniencing voters who have to vote twice
- Instant runoff voting.
In instant runoff voting, voters go to the polls just once and list not only their first choice but the second or even third or fourth choices.
The candidate with the least first choice voters is eliminated and that candidate's votes go to second choice candidates who are still in the running, and so on, until someone has a majority of the vote.
This eliminates the costs of a second round of voting, in exchange for making the initial ballot more complex and denying voters a chance of additional deliberation if their first choice candidate doesn't win.
- Single open primary voting.
In this system, everyone votes for their choice of a candidate and the top two candidates enter a second round, even if one candidate gets a majority in the first round.
This is similar to majority voting, but allows additional deliberation about the two most viable candidates even if one of them wins a majority in the first round, on the ground that, for example, everyone may be familiar with an incumbent, but not with a number of challengers, and that once the number of challengers is narrowed to one and people have a chance to examine the challenger more carefully, they may change their minds.
- Partisan primary voting.
In this system, the main ideological divide is already known and political parties prevent spoilers that plague.
Three Rounds
You could have three rounds until a majority is achieved and there is virtue in not necessarily only picking the top two finishers in a majority vote system if the runners up are close to each other. But if could cost more money and at some point if the people eliminated in the first round aren't sufficient to give remaining candidates a majority, you'd want some way to end it. But, having a second round of the top two plus any other candidate with say 90% of the vote of the runner up, if no one receives a majority and making a third round consist of only the top two candidates in the second round, isn't necessarily a bad rule if it is a close race between ideologically similar runners up. If the Republican primary in the U.S. in 2016 had had something like a three round system, someone other than Trump probably would have been the GOP nominee because the competition among multiple "establishment" candidates would have thinned out.