Cobra Kai is a karate dojo originally established by Terry Silver and John Kreese. Silver bought it for Kreese, his Vietnam War friend. In turn, Silver and Kreese had previously trained together under Master Kim Sun-Yung in South Korea.
In the Netflix adaptation we learn a lot more of the the ways of Cobra Kai as head student Johnny Lawrence reopens the Cobra Kai dojo to become Sensei 33 years later in an effort to redeem himself and get his life back on track, which in turn reignites the rivalry between himself and Daniel Larusso.
Cobra Kai Never Dies.