In elections that use the single transferable vote (STV) method, quotas are used (a) for the determination of candidates considered elected; and (b) for the calculation of surplus votes to be redistributed. Two quotas in common use are the Hare quota and the Droop quota. The largest remainder method of party-list proportional representation can also use Hare quotas or Droop quotas.
Under most STV systems it is possible to be elected by less than quota (at the end when the field of candidates is thinned to the number of remaining open seats) if not every ballot ranks all candidates, but the quota used (whether Hare or Droop) is the amount that guarantees election. A particular type of STV uses a "uniform quota" where anyone who exceeds that quota is elected. New York City used such a system during its use of STV in city elections. Its set quota was 75,000 and each borough got a number of members that depended on the voter turnout in the borough. One problem with the system was that the size of the city council varied from election to election depending on the voter turnout. (1.7M votes were cast and generally 20 to 27 councillors were elected.)
The quota sets the number of surplus votes held by elected candidates. For fairness it is important that the number of votes held by the early successful candidate after surplus votes are transferred away is not surpassed later by another candidate who is unable to take a seat as they are already filled. That is the reason why the quota needs to be set at a point high enough to prevent more candidates achieving it than the total number of seats.
General comparison
The earliest versions of STV used the Hare quota. The Hare quota is equal to the total valid poll (V) divided by the total number of seats (n), or V / n. The use of the Hare implies either there will not be any exhausted votes or one or more candidates must win with less than quota.
The Droop quota is smaller than the Hare quota, and was first suggested because it is the smallest quota that ensures that the number of candidates who reach the quota will not be greater than the number of seats to be filled. The Hare quota also does this but is larger than it absolutely needs to be. Any quota smaller than the Droop quota carries a real, or at least theoretical, risk of more candidates being eligible for election than there are seats to be filled. The Droop quota is the next integer larger than V / (n+1).
The difference between the two quotas comes down to what the quota implies. Winners elected under a Hare system represent that proportion of the electorate; winners under a Droop system were elected by that proportion of the electorate.
In a ranked-vote election in which there is only one seat to be filled (in other words an Instant Run-off Voting election) it is possible to use the Hare quota, which will simply be equal to 100% of votes cast. However, likely no candidate will take the Hare quota (due to exhausted votes) and such a target would necessitate vote transfers from eliminated candidates perhaps long after a candidate has a majority of the votes. it is more efficient to use the Droop quota, which will be equal to a simple majority of votes cast, meaning 50% plus one. In single-winner elections, receiving a majority of the votes is considered to be a satisfactory marker of rightful success, and either Hare or Droop would achieve that result.
Election systems that use quota produce what is thought to be more fair results, where the candidate(s) represent the majority of votes cast, whether with a single winner or multiple winners. Systems that do not use quotas do not ensure such fairness - often under First-past-the-post voting and the Single non-transferable vote system the elected candidate(s) do not reflect the majority of votes cast.
In an STV election, where there are multiple winners, quota, not plurality, is used because it ensures that most successful candidates are elected with the same number of votes. If they receive surplus votes or above the quota, the surplus votes are re-allocated. This equality ensure that a high proportion of votes cast are used effectively to elect someone, which in turn ensures that each party receives its due proportion of the seats.
Being of different sizes, the Hare and Droop quota produce different results in some situations.