The film is partially autobiographical. When Hayao Miyazaki and his brothers were children, his mother suffered from spinal tuberculosis for nine years, and spent much of her time hospitalized. It is implied, yet never revealed in the film, that Satsuki and Mei's mother also suffers from tuberculosis. He once said the film would have been too painful for him to make if the two protagonists were boys instead of girls.
The forest creatures and title characters of this movie got their name when Mei, the little girl who first sees them in the film, mispronounces the word "troll". At one point in the original Japanese language version, when Satsuki first finds Mei sleeping in the grove behind their house, Mei tells her sister she saw a "totoro". Satsuki replies, "Totoro, do you mean troll, from the storybook?" and Mei nods in agreement. This aspect of the story was left out of the 1993 Fox English version, probably because the difference between ""to-ro-ru" (the Japanese pronunciation of "troll") and "to-to-ro" would have been lost on English-speaking audiences. The quote is included in the 2006 Disney English version.
Hayao Miyazaki originally conceived the characters Satsuki and Mei as a single girl. He wanted to add suspense to the latter half of the film, and he felt it wouldn't work with just a single girl, so he split her into two separate girls. The original girl had features of both Satsuki and Mei, and was 7: halfway between the ages of Satsuki (10) and Mei (4).
Outside of the Japanese Academy Awards, which historically ignored animated films until 1998's Princess Mononoke (1997), "My Neighbor Totoro" won an award from every major Japanese film awards including Best Film from both the Mainichi Film Award and Kinema Junpo, as well as the Blue Ribbon's Special Award.
Totoro is actually a species name, as seen in the movie there are 3 Totoro creatures with different sizes and colors: the big gray one is Oh-Totoro (literally Big Totoro but in the English version he's called King Totoro), the middle-sized blue one is Chuu-Totoro (Medium Totoro) and the tiny white one is Chibi-Totoro (Little Totoro). Usually when people just say Totoro, they refer to the big gray Oh-Totoro.
The trivia items below may give away important plot points.
The sequence where Mei gets lost attempting to deliver an ear of corn to her mother includes her sitting by a row of statues. In Japan, such statues represent the Bodhisattva Jizo, the Buddhist deity who is the protector of children. Thus, the effect Miyasaki is conveying is a subtle reassurance to the audience that Mei, although lost at the moment, is otherwise in no immediate danger while her sister and King Totoro are coming to her aid in the Cat Bus.
Totoro actually comes from Mei mis-pronouncing Tororu, which means troll in Japanese. This comes from a book Mei had read, which turns out to be the Billy Goats Gruff, as can be seen in the end sequence which contains mother and that book. And if one looks closely, it is seen to have a totoro on the cover.