In the derelict ship, the crew of the Nostromo find the Space Jockey.
Apparently, he was infected by a Facehugger, so an Alien chest-buster had made his way out of his torso and bio suit.
What happened to this Alien?
This is most likely the original Xenomorph that laid the eggs that filled the spaceship and attacks the crew of the Nostromo in Alien and provides the aliens for the colony which appears in Aliens.
Of the alien itself, there is no mention, but it is reasonable to assume after laying eggs if it died, there was no reason for concern, the next generation of Xenomorphs was already underway. The eggs had shown remarkable environmental resistance and able to survive for millennia as they did in Aliens vs. Predator.
There is no reason to presume the Space Jockey was implanted on the ship, he may have been implanted at a previous location accounting for the lack of a facehugger corpse. It could have also been some time before he was discovered, it could have simply decayed.
I say that alien was alive and running around- but it happened to be far away from LV-426 when the Nostromo crew landed. It didn't see their shuttle come and land, and didn't know there was something it could hunt. Granted the alien likely would have been around when Hadley's Hope set up ship and started terraforming... but it's a big planet.