Beginnings
The art began with Mitsuyo Maeda (aka Conde Koma, or Count Combat in English), a Japanese expert judoka and member of the Kodokan. Maeda was one of five of the Kodokan's top groundwork experts that Kano sent overseas to spread his Judo to the world. Maeda left Japan in 1904 and visited a number of countries giving "jiu-do" demonstrations and accepting challenges from wrestlers, boxers, savate fighters and various other martial artists before eventually arriving in Brazil on November 14, 1914.
Since its inception, judo was separated from jujutsu in its goals, philosophy, and training regime. Although there was great rivalry among jujutsu teachers, this was more than just Kano Jigoro's ambition to clearly individualize his art. To Kano, judo wasn't solely a martial art: it was also a sport, a method for promoting physical fitness and building character in young people, and, ultimately, a way (Tao) of life.
The Name
When Maeda left Japan, Judo was also known as Kano Jiu-Jitsu and, even more generically, simply as Jiu-Jitsu. Teachers of both arts didn't try too hard to make the distinction clear. For example, Tsunejiro Tomita himself co-authored a book called Judo: The Modern School of Jiu-Jitsu in around 1906. Outside Japan, however, this distinction wasn't even hinted. Both arts, jujutsu and judo, were practically unknown. To the extent that they were known, they were considered the same thing. The distinction between a jutsu and a do is subtle, and is still used somewhat arbitrarily to this day. Thus, when Maeda and Satake arrived in Brazil, every newspaper announced jiu-jitsu despite both men being Kodokan Judoka.
The Japanese government itself did not officially mandate until 1925 that the correct name for the martial art taught in the Japanese public schools should be "judo" rather than "jujutsu". In Brazil, the art is still called "Jiu-Jitsu". When the Gracies went to the United States to spread their art, the system became known as "Brazilian Jiu-Jitsu" and "Gracie Jiu-Jitsu."
"Jiu-Jitsu" was also the original spelling of the art in the West and that is why this style retains the original (although technically incorrect) spelling. Other common spellings are Jujitsu, Ju-Jitsu, Ju jitsu and Jujutsu - the last being correct in accordance with modern romanization.
The art is sometimes referred to as Gracie Jiu-Jitsu (GJJ), but this name is trademarked by Rorion Gracie and specifically refers to the style taught by him and his selected teachers. Other members of the Gracie family often call their style by personalized names, such as Charles Gracie Jiu-Jitsu or Renzo Gracie Jiu-Jitsu, and similarly, the Machado brothers call their style Machado Jiu-Jitsu (MJJ). While each style and its instructors have their own unique aspects, they are all basic variations of Brazilian Jiu-Jitsu.
Development
Maeda met an influential businessman named Gastão Gracie who helped him get established. In 1917, his son Carlos Gracie, still a 14 year-old boy, watched a demonstration by Maeda at the Teatro da Paz and decided to learn jiu-jitsu. Maeda accepted Carlos as a student, and Carlos went on to become a great exponent of the art and ultimately, with his younger brother Hélio Gracie became the founder of Gracie Jiu-Jitsu, modern Brazilian Jiu-Jitsu.
In 1921, Gastão Gracie and his family moved to Rio de Janeiro. Carlos, then 17 years old, passed Maeda's teachings on to his brothers Osvaldo, Gastão and Jorge. Hélio was too young and sick at that time to learn the art, and due to medical imposition was prohibited to take part in the training sessions. Despite that, Hélio learned jiu-jitsu by watching his brothers. He eventually overcame his health problems and is now considered by many as the founder of Brazilian Jiu-Jitsu (though others, such as Carlson Gracie, have pointed to Carlos as the founder of the art).
Hélio competed in several submission judo competitions which mostly ended in a draw. One defeat (in 1951) came by Masahiko Kimura, whose surname the Gracies gave to the arm lock used to defeat Hélio.
The Gracie family continued to develop the system throughout the 20th century, often fighting vale tudo matches (precursors to modern MMA), during which it increased its focus on ground fighting and refined its techniques.