The Company is the corporation that employs the crew of the Nostromo. It is only ever called "the Company" by the characters, meaning it must have a seriously poor marketing department. That name just does not lend itself well to brand recognition. (Can you imagine the SEO problems?)
Symbolically though, the name is a little more useful, since it gives the Company a seriously creepy omnipresent quality. Like Big Brother, you can easily imagine a company called simply "the Company" invading every part aspect of the lives of future people.
In the film, the Company sends the crew of the Nostromo to the unexplored planetoid, hoping that one of them will become infected. It also instructs Ash and Mother to protect the creature as opposed to the crew. We're never given a reason why it would want the alien although Ripley guesses it is for their bio-weapons division. Did the Raccoon City incident teach these guys nothing?
As a symbol, the Company eerily parallels the alien. The alien preys on people to meet its own needs—food, safety, reproduction, etc—just as the company preys on people to meet its needs—that is, workers, materials, and profit.
When Ripley discovers the plot, Parker asks the busted-up Ash, "That damn Company! What about our lives you son of a b***h?" To which Ash coolly replies, "I repeat: All other priorities are rescinded" (Alien).
Clearly, the company has done the math, made a spreadsheet, and determined that the alien creature is worth more to their bottom line than the lives of six people. The Company's actions are unclouded by "conscience," "remorse," and "delusions of morality," the very qualities Ash admires in the alien. The alien may be an organism perfectly designed for survival of the fittest in the wilderness of space, but the Company is an abstraction perfectly designed for survival in the jungle of Wall Street.
Greed is Gory
The Company plays on the trope of the evil corporation that's all over film from the 1970s onward—think OCP from RoboCop, Soylent Corporation from Soylent Green, and Gordon Gekko from Wall Street. Beholden only to stockholders and their bottom lines, these corporations are the imperialists for the new era. They use people like tools and raw materials, killing and disenfranchising the poorest among us to develop a killer product or turn a profit.
This theme isn't fully developed in Alien, but if it gets your blood stirring, make sure to check out Alien's sequels. Ridley Scott also explores the issues further in Blade Runner, with the Tyrell Corporation.