Roger is a member of the cleaning crew onboard the scientific spaceship "Arcada", which holds a powerful experimental device called the "Star Generator". Roger emerges from an on-duty nap in a broom closet to find the ship has been boarded and seized by the sinister Sariens. Using a keycard he found from the dead body of a guy named Jerry, he finds his way to an escape pod and escapes the "Arcada". After crash-landing, he finds himself on a dry and barren wasteland that is the planet Kerona. Making his way through the desert and through a system of caves below, he is tasked with the killing of a creature called the Orat. Luring a Sarien Spider Droid sent by the aliens to find him, he leads it into the Orat's cave. It attacks the Orat and detonates, destroying itself and the Orat.
After succeeding in this task, he returns to the area where he had been given the task and shows the alien that assigned the task to Roger a piece of the Orat. He lets him through a door, and Roger is given a skimmer and the keys.
After navigating a rocky section of the planet, he goes to a bar, and, playing a game of slots, wins a sufficient amount of money so that he can buy a spaceship and a navigation droid. He does so, and flies to the coordinates (which he overheard from a customer at the bar) where the "Deltaur", the Sarien ship that boarded the "Arcada", is located. After getting aboard it, he infiltrates the "Deltaur" and self-destructs the Star Generator.
At the end of the game his efforts are rewarded when Roger receives the Golden Mop as a token of eternal gratitude from the people of Xenon and becomes an instant celebrity.
(from the Wikipedia entry for Space Quest.)
Space Quest: The Sarien Encounter is copyright Sierra On-Line (now Vivendi). This site provides an illustrated walkthrough, not the original game.