Halo explained SPOILER WARNING!!

The story begins with Captain Jacob Keyes, whose ship, the Pillar of Autumn, just made a blind jump into faster-than-light travel, called Slipspace. They were escaping the bombardment of the human colony Reach by the villains, a religious alien race called the Covenant whose sole goal is to exterminate the human race in a holy vendetta.
The Autumn is now near a gas giant called Threshold and its moon, Basis. They have found a large ring-shaped structure near it. Keyes is conversing with Cortana, a foot tall purple holographic woman who is the ship's AI and has control over all of its functions. Unfortunately, it seems that the Covenant have caught up with the ship. Keyes orders initiation of the Cole Protocol, a safeguard designed to lead the Covenant away from Earth should a human ship become a captive. It involves purging all evidence of Earth and her colonies from the ship's mainframe, and if it is captured the ship automatically self-destructs.
Keyes also orders all hands to get suited for combat in case of boarders. Meanwhile, in the ship's cryo bay, a pair of personnel get orders from Cortana to "unseal the Hushed Casket". That is where the Master Chief, the playable character in the game, lies in cryo-sleep. The last of a failed experiment to create super-soldiers codenamed SPARTANs, the Master Chief was kidnapped by the military at six years old and has undergone brutal training and surgical augmentation giving him heightened strength, intelligence, speed, and bone hardness.
The Chief is awakened while the Covenant are boarding the ship. He goes to the bridge and meets with Captain Keyes, who puts Cortana into his Mark V MJOLNIR armor. The Chief leaves the Autumn on an escape pod while Keyes attempts to land the ship on the ring structure.
The Chief also lands and rescues the surviving Marines via dropship pickup. However, Captain Keyes is captured by the Covenant and is taken prisoner aboard a Covenant cruiser known as The Truth and Reconciliation.
The Chief, along with a strike force of Marines, rescue him. While in captivity, Keyes recalls a group of guards discussing the ring. He heard that it was known as Halo and was presumably a weapon with ultimate power. Keyes tells the Chief to try to find Halo's Control Center before the Covenant, as they might use Halo to wipe out the human race.
Cortana hacks into the Covenant battle-net where she learns the location of The Silent Cartographer, a map room that points the way to the Control Room. The Chief and the Marines find the map room, and make their way to the Control Room.
Meanwhile, Keyes trys to find a Covenant weapons cache that could be useful for the war effort. While at the Control Room, Cortana tells the Chief to stop Keyes from finding the cache without a clear reason. It turns out that Keyes and the Marines accidentally released a race of parasitic life-forms known as the Flood, which reproduce using small infection forms that borrow into your body and tap into your spinal cord, turning you into a Flood form.
Chief finds a video recording proving all of it. He eventually escapes the cache and runs into a small floating blue robot who introduces himself as 343 Guilty Spark, who is the Monitor of Installation 04, presumably Halo. He says that Halo is supposed to completely annihilate the Flood, and, for reasons unknown, calls the Master Chief a Reclaimer. He teleports the Chief to a facility called the Library where he is supposed to find the Index, the key for activating Halo. The Chief fights his way to it and is teleported to the Control Room. Before the Monitor can activate Halo, Cortana, who is still inside Halo's systems, purges the Index form the system and stores it in her memory core. She reveals to the Chief that Halo doesn't kill Flood: it kills their food. That's humans, Covenant, all sentient life. The Monitor lied to him.
Now the Monitor and his Sentinels, floating robots with laser beams, are trying to stop the Chief from stopping the Monitor, who has found a secondary way to activate the ring. Fortunately, the Chief and Cortana do stop the Monitor. Cortana finds out that, once again, Captain Keyes has been captured and is held aboard the Truth and Reconciliation, except this time he is being assimilated into the Flood. Cortana teleports the Chief onto the ship, which is almost completely Flood-infected, where Keyes dies.
However, the Chief is able to remove Keyes' neural implants, allowing him to initiate the Pillar of Autumn's self-destruct sequence, which is halted by the Monitor. Needless to say, the Autumn is also Flood-infected. The Chief goes to the Engineering bay where he manages to re-initiate the self-destruct sequence, allowing him to escape just as Halo, the Flood, and the Covenant armada still on the ring are completely obliterated.
Coming up: Halo 2 explained.