Beware, many "Prometheus" spoilers below.
"ALIEN: COVENANT'S" PLACE IN THE TIMELINE
"Alien: Covenant" is set between the events of "Prometheus" and "Alien" and is directly informed by the events of "Prometheus," which took place only 10 years earlier. While a re-watch of "Alien" isn't crucial for "Covenant," there are some interesting parallels that might be fun to have fresh in your mind.
WHAT WAS "PROMETHEUS" ABOUT?
A megalomaniac billionaire, Peter Weyland (Guy Pearce), who created the android David (Michael Fassbender), funds an expensive mission to a distant moon to explore a theory posed by two scientists that humans were created by other beings. The scientists, Elizabeth Shaw (Noomi Rapace) and Charlie Holloway (Logan Marshall-Green), call them the Engineers.
They find what they're looking for (though most are long dead). Prometheus' crew soon discovers that while the Engineers may have created the human race, they also at some point "changed their minds" and became architects of destruction.
WAIT, WHO ARE THE ENGINEERS?
Giant, Adonis-like, milky-skinned and hairless beings who ostensibly created humans and then tried to kill them with a pathogen that causes various mutations. One Engineer is still living in "Prometheus" and David wakes him up.
THE ORIGINS OF THE ALIEN (AKA XENOMORPH)
Charlie ingests the pathogen (David slips the black, milky substance into his drink) and then gets intimate with Elizabeth Shaw, who, the next day gives birth to a trilobite, which is basically a giant Facehugger. It attacks the lone living Engineer, who then births a proto-Xenomorph. It's a variation of the "good old beast," as Scott calls it, who sprung out of John Hurt's chest and terrorized the Nostromo crew in "Alien." You'll meet a new iteration, and some other creatures, in "Covenant" too.
This is a big one. The remaining Engineer on the planet attempts to take off in his cornucopia-shaped spacecraft, but is stopped when Idris Elba and his mates crash the Prometheus into his, killing everyone aboard.
On the ground, the only survivors are Elizabeth and David, who's been decapitated. David convinces Elizabeth to help him — he knows how to operate other ships on the planet and is her only chance for survival. But Elizabeth doesn't want to go back to Earth. She wants to find out where the Engineers are from. The last scene shows another spacecraft flying away, seemingly with Elizabeth and David on board.
THE IMPORTANCE OF DAVID
Fassbender's David plays a key role in "Alien: Covenant," and it's useful to remember that he, like the android Ash (Ian Holm) in "Alien," helps trigger most of the disasters in "Prometheus." He's outwardly disappointed to discover that the Engineers are mortal, and is much more interested in the pathogen and what it can create. Fassbender also plays another synthetic, Walter, who works on the Covenant.
THE OTHER "ALIEN" MOVIES
For the purposes of "Alien: Covenant," you're safe to disregard the three "Alien" sequels, and the "Alien vs. Predator" movies, none of which were directed by Scott.