As an example of the type of men the Golden Company attracts, Jorah Mormont revealed in Season 4 that as a disgraced knight in exile, he fought with them before signing up with Viserys and Dany. The Golden Company’s captain-general is Harry Strickland, who grew up in the Company, as did his father and grandfather after his great-grandfather signed up with the original outfit.
That speaks to another distinguishing feature for this squad: They raise their sons to join them. It’s part of their heritage.
The Golden Company was founded by a legitimized Targaryen bastard — Ser Aegor Rivers, known as Bittersteel — as a fighting force to support rebellions against the Iron Throne. Bittersteel’s dad, King Aegon IV, known as Aegon the Unworthy, had a whole litter of bastards that he legitimized on his deathbed a century ago. This led to a series of succession clashes known as the Blackfyre Rebellions, in which Aegor Rivers and the Golden Company supported the Blackfyre claims. This is why some see them as anti-Targaryen.
In the books, Dany’s brother, Viserys, tried to hire them once, and got laughed off. This might not have been simply because Viserys was a Targaryen: Later, on the Company supports a separate Targaryen claim to the throne and invades Westeros after breaking a contract for the first time in order to do so.
By Season 4, when Davos recommends them, the Golden Company has 10,000 men: 8,000 infantry, 1,000 archers and 1,000 cavalry. They also have two dozen war elephants. Their infantry are phalanx-style spearmen, like the Unsullied, but with more adaptability. Their archers have a variety of weapons and styles — crossbows, recurve-bows, longbows, and mounted archers. Together, they are a much better match than a Westerosi-trained army would be against the combined forces of an all-cavalry Dothraki and an 8,000-strong all-infantry Unsullied.