The Unicorn is one of the monsters seen on the whiteboard. It is shown that the Engineering Department bet on it.
Physical Description & Attributes
It's shown as a white horse with a large horn on its head.
It is seen during the Purge when it chases and stabs a worker with its horn.
There is Unicorn Tapestry in the basement, if it is tampered with it will summon the unicorn.
Inspiration
Unicorns have appeared in horror films before,however, maybe Drew Goddard and Joss Whedon put it as a inside joke and/or an extreme sarcastic note on how many other mythical creatures have been used in filmmaking as evil ones. Some people have argued that maybe this is a metafiction note on the fakeness of the scenario, linking it to the famous origami unicorn in Blade Runner.
Unicorns, despite their majestic and approachable nature in today's fiction, are actually known in lore as being deceptively violent in nature. Traditionally, a unicorn would be drawn to the purity of a virgin, but if one was merely pretending to be a virgin a unicorn would actually kill that person. Inside joke or not, there's a lot of knowledge and backing to this little addition.