Condorcet is an ordinal voting system that is believed by its supporters to be fairer than and superior to the plurality vote, instant runoff voting, the Borda count, approval voting, and all other voting systems which select a single candidate. It is based on the ideas of the Marquis de Condorcet, who stated that in an ideal voting system, a candidate who is preferred by a majority of voters should always be elected.
Each voter ranks the candidates in order of preference. Any candidates not listed are assumed to be ranked equal to each other and below all listed candidates. A Condorcet winner is said to exist if a given candidate is preferred over every other candidate by a majority of voters. When a Condorcet winner exists, he or she is elected. Otherwise, the Smith Set is found - the smallest set of candidates that is preferred over all other candidates. Various contingency plans exist for selecting a candidate from within the Smith Set, but I will not discuss those here.
Here's how it works: Suppose we have three candidates: A, B, and C. Twelve people voted A, then B, then C. 8 Voted A, C, B; 6 voted B, A, C; 17 voted B, C, A; 12 voted C, A, B; and 23 voted C, B, A. This can be written as follows:
12 A B C
8 A C B
6 B A C
17 B C A
12 C A B
23 C B A
Now a 3*3 matrix is constructed. The number given is how many people prefer the candidate specified by the row over the candidate specified by the column.
A B C
A - 32 26
B 46 - 35
C 52 43 -
We see that 46 people prefer B over A, and only 32 prefer A over B, so B is preferred over A by 14 votes. With three candidates, there are only three pairs, which are as follows:
B > A: 14
C > A: 26
C > B: 8
C is preferred over both A and B, so C is the Condorcet winner.
Perhaps the biggest problem with Condorcet voting methods is their complexity. In addition the problems of replacing all existing voting equipment, implementation of such voting methods is unlikely because people will tend to be wary of a system so counterintuitive.
Supporters of Condorcet methods are particularly disdainful of Instant Runoff Voting, and with good reason. IRV is very difficult to implement from a technical perspective, because a lot more data has to be stored, and because its results are considered to be inferior. Nevertheless, it seems to have the jump on Condorcet. IRV has many supporters because it is easy to understand.