The "Watcher" ship is a purple UFO which can only be acquired in a few places in the game.
The first opportunity to get it is area 4-6. The loading screen before this section will mention that there is something jumping in and out of their radar. This is the UFO.
Two sets of ships which fire triple lasers will appear. Sit on the bottom or top of the screen at the start of the wave and destroy those two ships. The UFO will appear as a purple ship skipping along the top and bottom of the screen, teleporting back and forth between them.
Jump to it, and you will gain this achievement.
The UFO is a flying saucer which fires forwards and backwards, and can wrap around the edges of the screen. It has an infinite amount of ammo, but if you get hit even once while piloting it, you die.
Another flying saucer appears in 7-8, and is much more useful for unlocking the Gone Home achievement, which requires you to pilot that saucer all the way through to the ending of the game.