"I work at a company." - the speaker might be relating the location of the company he's / she's working with.
"I work for a company." - here, the speaker is suggesting that he/she is a part (employee) of the company.
the statements however are gramatically correct although, most of the time, you would likely hear the specific company name they are working with:
"I work FOR McDonald's (for instance)."
or, unspecifically,
"I work FOR a well known car manufacturing company."
:)