The training video above explains the basic premise: in 2176 Jien Garson (voied by Sense8's Jamie Clayton) convinces Earth and several other worlds to sponsor the Andromeda Initiative. This project sees three large colony ships and an even larger mobile base, the Nexus, constructed with the goal of colonising the Andromeda galaxy, over two million light-years from the Milky Way. In 2185, between the events of Mass Effect 2 and 3, the Initiative is launched. All four vessels are well outside the Milky Way galaxy when the events that conclude Mass Effect 3 take place, meaning that they are not effected by the choices the player makes at the end of the trilogy.
Andromeda itself will begin circa 2780. After a journey of roughly six hundred yeaers in cryosleep, the Nexus and the colony ships have reached the Helios Cluster, a region on the outskirts of the Andromeda galaxy. Unfortunately, one of the ships, the Hyperion, crewed and operated by humans, has gone off-course. You control either Scott or Sara Ryder, the children of Alec Ryder, the Hyperion's Pathfinder. Something happens and Alec goes AWOL, leading you to pick either Scott or Sara (the other character remains in the game as an advisor, but not a squad-mate) and promote them to replacement Pathfinder. Unlike Shepard from the first three games, who was already a combat veteran, your character in Andromeda will be untested and new to command.
The game will unfold in a broadly similar manner to the previous trilogy, with you flying to new worlds and taking part in missions. This time there's more freeform content based on exploring each planet in your new vehicle, the Nomad (an advancement from the Mako), and getting into scrapes with new aliens. There is a new adversary race, the Kett, but not much has been revealed about them. BioWare want to keep a lot of surprises for the game itself, which sounds like it will be focused much more on exploration than the previous games in the series. However, they have revealed a new classless levelling system, based instead around skills, and a more nuanced take on morality rather than the old binary Paragon/Renegade system.
Mass Effect: Andromeda will be released in Spring 2017 on PC, X-Box One and PlayStation 4.