Parliamentary Debate is a partnered debate event featuring two sides to the argument; one being the government and the other opposition. The government supports the motion (topic of debate), discussing a specific issue and solution in the topic area. The other side, the opposition, finds weaknesses in the government's argument. Parliamentary Debates discuss current issues, government laws, and other relevant topics.
Video #1: Sample HS parli debate competition
Video #2: Parliamentary debate explained
Link: National Parliamentary Debate League website
Congressional Debate is a mock legislative assembly competition where students draft bills (proposed laws) and resolutions (position statements), which they and their peers later debate and vote to pass into law and then take action on by voting for or against the legislation. (https://www.speechanddebate.org/wp-content/uploads/2017-Congressional-Debate-Guide.pdf)
Public Forum Debate features four high school students on teams of two debating a timely issue in highly structured speech times. The teams compete for the vote of a judge or panel of judges, who will decide the round based on which team debated better. (https://www.speechanddebate.org/wp-content/uploads/Public-Forum-and-Congressional-Debate-Textbook.pdf)
A two-on-two debate that focuses on a policy question for the duration of the academic year, this format tests a student’s research, analytical, and delivery skills. Policy Debate involves the proposal of a plan by the affirmative team to enact a policy, while the negative team offers reasons to reject that proposal. Throughout the debate, students have the opportunity to cross-examine one another. A judge or panel of judges determines the winner based on the arguments presented. (https://www.speechanddebate.org/wp-content/uploads/MS-Policy-Guide.pdf)
Only Public Forum, Policy, and Congressional debate are through the NSDA. (Parliamentary Debate is a regional debate style and not through the NSDA. People debating in that style might not be able to compete at all tournaments, namely at UPenn and Districts.)