Do mean Condorcet's paradox, or Downs paradox? Both are referred to as the voting paradox sometimes.
Condorcet's paradox is basically that, in a preferential ballot (a vote where voters rank candidates in order, rather than just selecting one) it is possible for there to be no candidate who is preferred by a majority of voters.
Say there are three candidates (A, B and C), and thirty voters. Ten voters put A first, B second, C third. Ten voters put B first, C second and A third. And ten voters put C first, A second and B third. If any candidate wins, it can be argued that two-thirds of voters would have preferred a different candidate. If A wins, the second and third groups of voters would have preferred C. If C wins, the first and second groups would have preferred B. If B wins, the first and third groups would have preferred A.
Downs paradox is essentially saying that, except in the extremely unlikely event that your preferred candidate wins by a single vote, your own participation in an election is inconsequential. Because of this, the costs of voting (the time and effort involved) should persuade a rational, self-interested voter to not vote. The costs will outweigh the benefits, since it is very unlikely that there will be any benefit at all. Thus the fact that people actually do vote in elections, is considered paradoxical by those who regard people as motivated by rational self-interest.
The paradox is avoided if voting is compulsory, or if you consider that the voter benefits from having expressed him or herself, even if theirs is not the pivotal vote.