After the aliens on the bridge have been defeated, you should head over to the Captain's chair. An alien will appear and be very upset, along with another mothership. Activate the right two buttons to raise the shield.

After the alien fires on you, activate the left two buttons to activate the Deathray. Fire using the big control button in front of you, then raise the shield again. Repeat until the alien ship is destroyed.

Part way through the battle you'll lose power. There are two controls on each side of the bridge. Activate them to get power back. You'll also have a few aliens invade the bridge. Take them out and give stimpacks to your team by talking to them. The gunfighter takes the worst of it.

When the alien ship is destroyed you'll get the quest complete notification. Talk to everyone and feel free to look around. It's finally over.

When you are ready to leave, activate the teleporter to return to the Capital Wasteland. Sally and Elliott will remain on the ship.

This beacon will allow you to return to the ship anytime, so that you can continue looting at your leisure. It's also nice to finally see some color in the world of Fallout, though it is ironic that it's mainly aboard an alien ship.