In an open primary, voters of any affiliation may vote in the primary of any party. They cannot vote in more than one party's primary, although that prohibition can be difficult to enforce in the event a party has a primary runoff election. In many open primary states, voters do not indicate partisan affiliation when they register to vote.
One area of contention in open primaries is "crossover" voting. It most often involves voters affiliated with one political party voting in the primary of another political party to influence that party's nomination. For example, if a district routinely elects the Democratic nominee, Republican voters may vote in the Democratic party primary to attempt to influence the outcome. This could be a good-faith attempt to select a more conservative Democratic nominee who would be palatable to the Republican voters, or it could be sabotage, an attempt to nominate a weaker candidate who is easier to defeat in the general election.
In a closed primary, only voters registered with a given party can vote in that party's primary. States with closed primaries include party affiliation in voter registration so that the state has an official record of what party each voter is registered as.
Closed primaries preserve a party's freedom of association by better ensuring that only members of the party influence that party's nominees, but critics claim that closed primaries can exacerbate the radicalization that often occurs at the primary stage, when candidates must cater to their party's "base" rather than the political center.
In a few states, independent voters may register with a party on Election Day. However, they must remain registered with that party until they change their affiliation again. A handful of states even allow voters registered with one party to switch their registration at the polls to vote in another party's primary. In these rare instances, a closed primary can more closely resemble open or semi-closed primaries than the closed primaries of other states.
In a semi-closed primary, unaffiliated voters may choose which party primary to vote in, while voters registered with a party may only vote in that party's primary. Representing a middle ground between the exclusion of independent voters in a closed primary and the free-for-all of open primaries, the semi-closed primary eliminates concerns about voters registered in other parties from "raiding" another party's nominating contest.
People who align with a given party may theoretically still vote in another party's primary if they are registered as independent. The potential for such tactical party registration is also present in the strictest of closed primaries.