The Triad is a blanket term for the real life criminal organisations formed in 18th century China, now with factions all over the world. The Sun Yee On Triad, founded in 1919, are based in Hong Kong are one of the largest and most powerful Triads. It is to this Triad Chang and the Triad members we see in Black Lagoon are said to belong. Within the manga and the anime Mr Chang's organisation is simply referred to as "The Triad."
The Triad is structured so that each branch has a "boss," sometimes referred to as "big brother," or "dailo," with members beneath them, often referred to as "little brothers."
Individual members of the Triad are granted a large degree of autonomy, provided they follow the 36 oaths which Triad members take upon joining. All the oaths are essentially some variation of promising loyalty to the Triad and it's members, or else face death by swords/thunderbolts/etc.
The Triad that we see in Black Lagoon are the Roanapur branch of the Sun Yee On Triad, of which Chang is the boss. Chang answers to the head of the Sun Yee On Triad in Hong Kong.The Triad make only a few major appearances. They first accompany Mr. Chang to the meeting with the Jihad terrorists. Later they are seen with him at the Black Lagoon crew's office. After that many of them are killed when Chang goes to confront Hansel and Gretel.
The Hong Kong Triad had fought against Balalaika and Hotel Moscow in '93 where both Chang and Balalaika were wounded by each other's guns. The Triad seem to have a lot of resources, as they were able to make a deal with agents from the CIA. Most of the members carry sub-machine guns. The only members seen without the full group are Biu Yuen and Chang.
There is only one other named Triad member, Cho, who was killed by the vampire twins along with another unnamed Triad member.The 14K Triad are also mentioned in the anime, as Chang says that Hansel and Gretel killed some of it's members.
In the anime, Chang refers to members of the 14K Triad as "their men," however in real life the Sun Yee On Triad and the 14K Triad are rivals. It may be the case that Rei Hiroe chose to name these two Triads as they are the two largest and most powerful Triads operating in Hong Kong, without realising they are rivals.
There was a real Hong Kong police officer who joined the Sun Yee On Triad, however unlike Chang he asked the cops for protection in 1983 (in what became known as the 1980's spill), and identified the head of the Triad, who was found guilty and sentenced to seven and a half years in prison.
There is an important but complicated distinction to be made between the Hong Kong Triad groups and those on mainland China.
Women are permitted to join the Triad, and some real-life Triad bosses have been women, however, the Triad members we see in Black Lagoon are all men.
The Sun Yee On Triad allegedly invests heavily in the Hong Kong film industry, which is why gangsters and Triad members are often romanticised in Hong Kong cinema. This could be why we have many famous characters played by Chow Yun Fat in John Woo films, who are said to have inspired the character of Mister Chang.