Yakuza (極道 Gokudō?) refers to Japanese organized crime syndicates and their members.
The Yakuza includes many Japanese crime syndicates that controlled the underworld in the past. At this time the Yakuza's influence and power were unrivaled as its members thrived as honorable gangsters. The Yakuza were notorious for their strict code of chivalry unlike today's Villains.
During the rise of All Might and the age of Heroes, the majority of Yakuza factions were systematically dismantled due to becoming recognized as Villain organizations. The time for the Yakuza as the rulers of darkness came to an end and most were locked away.
Any remaining Yakuza were treated as potential villains and lived under constant surveillance. In modern society, many consider the Yakuza to be small-time thugs on the brink of extinction. 
Following the collapse of the Todou Gang, the Shie Hassaikai reflect on the Yakuza's shame as one of it's only remaining groups. The Boss refuses to give up their strict code of ethics and raised Kai Chisaki to believe in them.
The Boss revered the disposition of the bygone era and tried to find a path where gangsters could thrive in the current world. This gave him a following that allowed the Shie Hassaikai to survive for some time. 
However, Chisaki strays from the path in order to return the Yakuza to prominence and transforms the Shie Hassaikai into a villainous organization.  The true Yakuza members only remain out of respect for the Boss and fear of Overhaul's tenacity.
- Yakuza - Wikipedia article about the real-life Japanese Yakuza.