There are many forms of dance called 'Swing'. The original form is called Lindy Hop, and was developed in the 1930s in Harlem, New York.
Swing peaked in the 'Swing Era', the 1940s, and evolved into various forms, including the 'Rock&Roll' dancing styles of the 50s and early 60s. It was finally overtaken in popularity by the non-paired dance styles of the 1960s, such as the twist, and freestyle dancing.
Swing had a giant revival in the 1990s. In the modern college 'swing' scene, the dancing is casual, fun, fast, and generally danced as a single-swing step pattern. In the ballroom circuit, though, a triple-stepped version (Triple Swing or Jive) is more common.
The video shows Charleston, from which swing evolved, then two forms of swing: Lindy Hop, and Balboa.
For different forms of swing, see: