After the incredibly slow build of the first two acts of the film, the Covenant crew are decimated by the alien pathogen in various ways (two are killed directly through exposure, others by the neomorphs who burst out of the first two - directly or indirectly) only to discover that their sanctuary in the Engineers' temple is a trap set by David.
When his secret is revealed, David reveals that he spent the time following the Prometheus incident "experimenting" with genetics - partly from the pathogen and partly using Elizabeth Shaw's oddly well-preserved DNA (she's dead, Jim). The results of his various experiments have eventually led to the development of the traditional Xenomorph eggs containing Face-Huggers, which he "feeds" with captain Oram (Billy Crudup).
He's infected, the xenomorph bursts out and kills almost everyone else before it is eventually killed by Daniels in a mid-air battle on the outside of a rescue ship and David is apparently killed by fellow android Walter. Naturally, there's a second Xenomorph inside one of the survivors, which then kills everyone aside from Daniels, Walter and Tennesee, who work together to kill it before it too is killed and the human survivors head into stasis.
Obviously, it turns out that Walter is in fact David in disguise, who adds two Face-Hugger embryos to the colony payload and continues on to the original colony planet.