USCM
The Weyland-Yutani facility has not been heard from in six weeks, and the U.S. Colonial Marines are to rescue survivors and secure the facility until the 6th Battalion arrives.
The dropship is blown off course by hurricane winds and damaged. Soon after, Sergeant Hall is killed by an Alien. The marines call for help, and must defend themselves against Aliens and Predators until help arrives. They manage to shut down the security grid, but upon finding the pods, realise they have endangered the survivors there in doing so. Not long after, the marines are betrayed by General Rykov and his Iron Bears to protect the secrets of The Weyland-Yutani facility. McCain and his team are sent on a wild goose chase into the heart of the alien hive, after being informed Harrison has been killed in an accident. Harrison is imprisoned and is to be killed, but escapes. He is contacted by a mysterious person and is sent to retrieve data about the facility and the experiments.
After escaping the pods, he learns his 'helping voice' is Tomiko, the sister of a deceased employee of the facility. She helps him make his way toward his team. Harrison stumbles across the other half of his team (the half who stayed at the drop point), but they are decimated by a predator who is stalking Harrison. He finally battles the predator and defeats it.
Harrison uses an exo-suit to battle his way into the hive and ultimately defeats the alien empress. He holds off the attacking alien drones as the drop ship lands and he and the other surviving marines make their escape. Tomiko sets off the self-destruct system as the dropship left the vicinity of the hive, having Major McCain advise USCM command that any further rescue and salvage attempts at the facility be abandoned.
Predator
A backstory is shown where a predator is hunting a lone soldier. The predator severely wounds the soldier but he is found alive by his team and is rescued.
Many years later, Predators arrive on LV-1201 for a 'hunt'. After they kill a few humans, the majority are captured by the scientists of the Weyland-Yutani facility. A lone remaining predator follows them back into their complex to free the others but is also captured. While in the scientists custody, the Predators are subjected to torturous experiments. Thanks to the inadvertent help of a marine (Harrison), the main predator escapes and kills all of the humans around him. He discoveres that one of the people, General Rykov, was once a target of his (from the backstory), so he sets off to find Rykov and kill him.
He finds Rykov encased in a battlesuit. The Predator is able to defeat him, and takes his spine as a trophy.
Alien
An Alien egg finds its way into the Weyland-Yutani facility. When it hatches, the facehugger finds an isolated guard asleep and attaches itself to him, impregnating him. When the Alien bursts from the guard's chest, it finds a cat to feed on and grows into a powerful alien drone. It begins to wreak havoc in the facility, killing anyone and anything it sees, and causing the situation that drew the marines to the facility in the human story.
Part way through the game, the scientists attempt to steal an Alien queen, in an attempt to control the aliens. The Alien rescues the queen and kills all who tried to steal her.
Final Cutscene
In a final cut scene (which is accessible after completing all three campaigns in no specific order), the USS Verloc is seen departing LV-1201 with the US Colonial Marines. As it heads into deep space, the Predators can be seen stalking the ship in their own sleek battleship, appearing to chase their next 'hunt'.
After completing all three campaigns, a cryptic message is displayed during the credits:
Remote transmission acquired. Boosting for outgoing dispatch. Routing now. ... ... ...
The mule has been acquired. Package is en route.
...
ETA: Unknown ...
Mission Complete.
Extraction required. Priority status.
It is implied that the "mule" was the data disk that Harrison took in the USCM plot.