The two most common single-winner systems in the United States are:

Plurality voting, where each voter casts a vote for one candidate and the candidate with the most votes wins, even if it's less than a majority.

Two round runoff, where each voter casts a vote for one candidate, and a runoff is held between the top two candidates if no candidate receives a majority (or some other threshold, such as 40%) of the votes cast.

Fairvote supports the use of instant runoff voting for single-seat offices in public elections. Voters rank candidates in order of choice, winning requires a majority of the votes, and candidates are successively eliminated and ballots are recounted if no candidate receives a majority in a round.

A Borda Count is the technical name for the voting system in which a first place vote is worth 4 points, a 2nd is worth 3 points, a 3rd 2 points and a 4th is worth 1 point. The candidate receiving the most points wins. The Borda Count is often used to rank sports teams or to induct athletes into halls of fame.  One problem with the Borda Count is that ranking a less preferred choice will count againt your favorite choice.

In Approval Voting, each voter can approve of (vote for) as many candidates as she supports. The candidate with the greatest approval wins. Approval voting only measures whether or not a candidate is acceptable to the voter; it does not distinguish between a candidate who is intensely liked - a first choice - and those who are more weakly approved of -- second and lower choices.  While simple in design, approval voting creates incentives for complex campaign strategies.
It also could result in the defeat of a candidate whom an absolute majority support as their first choice.

The Condorcet rule elects the candidate who can top each of the others in a series of head-to-head contests. If most voters prefer (rank) A over B, A wins that contest. The rankings are used to determine the winner of each possible head-to-head contest.  The Condorcet rules suffers from the Condorcet Paradox:  there may not be any candidate who defeats all the others:  A might beat B, B might beat C, and yet C could beat A.  In this case, some other system must be used to resolve the paradox. In addition, the Condorcet candidate might be one with so little core support that  he or she would never have been able to win under any of the single-winner voting systems currently used for all governmental elections in the United States and other nations.

[ Examples of Flaws in Voting Systems ]