An Equally Weighted Vote is the concept that every vote should carry equal power or weight. In 1964, Wesberry v. Sanders, The U.S. Supreme Court declared that equality of voting - one person, one vote - means that "the weight and worth of the citizens' votes as nearly as is practicable must be the same."
Votes can be unequally weighted at a number of different stages in the election process. First, a vote can be unequal due to the voting method itself. Any voting method which allows Vote Splitting ensures that voters do not have an equally weighted vote in elections which have more than two candidates. Second, votes for representatives to a larger geographical area who are representing a district within that area can be unequally weighted due to district lines which may bias an election in favor of one faction or another. When district lines are intentionally drawn in order to marginalize specific factions, (reducing the weight of those voters relative to others) it's known as Gerrymandering.
The Electoral College and other mechanisms which use representatives to determine elections rather than directly using the votes cast also violate the Equally Weighted Vote, particularly in cases where electors or representatives are not allocated proportionately to the population. In the case of the Electoral College each state is awarded electors based on the number of members of congress. The House of Representatives is based on population, which would ensure that electoral votes were equally weighted as nearly as is practicable, but each state is also awarded two additional electors per state corresponding to their two Senators. This results in US presidential elections which specifically violate the Equal Vote Criterion.
The 1964, Wesberry v. Sanderscase cited above addressed Gerrymandering. In the case of district lines it's impossible to ensure that elections will not favor one faction or the other over time as populations grow and change, but it is "practicable" to prevent and mitigate this phenomena. However in the case of vote splitting and the Electoral Collage achieving a perfectly Equally Weighted Vote is fully possible.
Equal Vote Criterion
Any voting method or election which passes the Test of Balance passes the Equal Vote Criterion and can be said to guarantee an Equally Weighted Vote.
The Test of Balance
The test of balance is defined as the following "Any way I vote, you should be able to vote in an equal and opposite fashion. Our votes should be able to cancel each other’s out."
Voting methods which ensure an Equally Weighted Vote
Voting Methods which ensure an Equally Weighted Vote with any number of candidates include Approval Voting, Score Voting, STAR Voting, as well as a number of others. In general Cardinal Voting methods ensure an Equally Weighted Vote for each voter.
Choose One Plurality Voting only satisfies the Equal Vote Criterion in elections with two candidates only. Instant Runoff Voting (often referred to as Ranked Choice Voting) does not satisfy and the Equal Vote Criterion.