The English title, "A Better Tomorrow", is the translated title of the song sung by the two school kids (in Mandarin) in the middle of the movie. It's a New Year's Eve song, usually sung by large groups.
The character of Mark Gor proved to be so popular that many young HKers adopted his style (black trenchcoat, sunglasses, match in mouth). Stores around HK were sold out of the distinctive Ray-Ban sunglasses in less than a week after the movie premiered. All this led to a blasting of Woo by critics and government officials that he was glamorizing the Triad lifestyle.
Some theatre owners originally expressed concern to producer Hark Tsui about the casting of Yun-Fat Chow. Even though he was popular on TV, most of his films had been flops. Chow's nickname in industry circles for many years was "box-office poison".
When it was released in the Philippines in 1989, it was retitled as Better Tomorrow: Rapid Fire II. Distributors promoted it as sequel to _Rapid Fire (1988)_ which became a hit in that Asian country in 1988. It has nothing to do with Rapid Fire.
The movie is actually a remake of a 1967 Cantonese film called Ying xiong ben se (1967) (Story of a Discharged Prisoner). Hark Tsui had been toying with the idea since his days in the TV business, but because of an overwhelming workload, had to pass the directorial reigns to John Woo.
The translation of some of the characters' names are incorrect in the English versions. Kit's proper Chinese name is Ah-Git, and while "Mark" is correct, "Mark Gor" is inaccurate as in a first and last name. "Gor" means "brother" in Cantonese hence Mark Gor is translated as Brother Mark in English.
During the nightclub scene, the song being played in the background is the Cantonese version of a classic South Korean song called 'Hee Na Ree' sung originally by Goo Chang-mo in 1985. The Cantonese version in the movie was sung by Roman Tam, considered the "godfather" of the musical genre Cantopop.
The scene in which Mark Lee tells the story of being forced to drink urine is apparently based on a real incident involving Yun-Fat Chow and director Ringo Lam, according to Bey Logan on the DVD commentary. This scene was recreated in Woo's Bullet in the Head (1990).