A fighting company, often shortened to company, is a group of people that have joined together to fight as a unit. Unlike households, of which a person can join many at a time, a given person may only be a member of a single company.
From the rulebook
The First Edition Rulebook states the following:
Companies and Armies: "People who are friends and/or those of a certain persona type may wish to band together. This group is called a company for general purposes. Each company must choose a name, a captain, a sergeant, and produce a flag or devise. The name should be something orginal and interesting, for instance, "The white wolves". A company of a single nationality or race, say orcs, might want the name "The Orcs". But as more ors show up, they would probably want to distinguish themselves (for example- "The Red- Hand Orcs" or "Slurth's Raiders".) A group of Elves, Danish Vikings, and wargs calling themselves "The Celts" is less creative and not reccomended. A company's interest would be preferrable (The Blue Mystics' Guild or "John the Brave's Free Company")."
Joining a company
"This is like picking a good pair of shoes, get the ones that fit right for you and save a lot of blisters." -Thoron Underhill
Each company has its own policies, its own style and culture. Take the time to get to know the group before you think about joining. Most people suggest that you avoid even thinking about joining a company until you are at least a year into your career.