The Jets and Patriots are in the same conference. The Giants are in the other conference. You are correct in stating that each conference champion plays in the Super Bowl. The American Football Conference (AFC) sent New England, and the National Football Conference (NFC) sent the Giants.
I don't know why the Cowboys are in the same division as the Giants, Eagles, and Redskins, who are all along the coast, while Dallas is in the center of the USA...and much further south than even Washington, DC. It would make more sense to have the Falcons or Buccaneers in with those three teams, and the Cowboys with the other NFC South teams.
Edit to add: here
is a list of teams in their respective divisions, made up of four teams to each division, and sixteen in each conference. The red teams are the AFC, and the blue are the NFC.