David Haller (Dan Stevens) is a powerful mutant telepath and telekinetic. Canonically, in the X-Men comics, he is the son of Professor Charles Xavier, but that's been only implied so far in this series.
In resolution of the Season 1 cliffhanger, David has returned to the mutant refuge Summerland and is using the lab there to telepathically search for his nemesis, the Shadow King. (If you're familiar with the X-Men, you can say Summerland's lab is like a jerry-rigged Cerebro: it boosts a telepath's abilities and projects them over long distances.)
The Shadow King is a purely telepathic entity that attaches itself like a parasite to other telepaths. It was attached to David for most of his life, causing him to suffer psychotic delusions that eventually put him in a psychiatric hospital, which is where we started Season 1. It is currently in the form of David's friend and fellow patient Lenny (Aubrey Plaza), although it can take the form of almost anything in order to hide in a host's mind.
With the help of Summerland, David learned to control his powers and expel the Shadow King from his mind, but it simply reattached itself to Oliver (Jemaine Clement), the founder of Summerland and another telepath who specializes in astral projection. Oliver, possessed by the Shadow King (perhaps willingly; Oliver is strange that way), escaped at the end of Season 1.
This dance battle is a metaphorical representation of the telepathic battle between David, Oliver, and the Shadow King.