If you haven't heard his name, you must have heard about at least one of his creations. So, who's Miyazaki? It's easier to say who he's not.
Hayao Miyazaki was born on the 5th of January 1941 in Bunkyo, Tokyo, as a second of four sons. His father was a director in young Miyazaki's uncle's factory producing rudders for fighter planes - this is where Hayao became highly interested in aviation, which often appears in his films.
When Miyazaki was a little kid, his mother suffered from Pott's disease (sort of tuberculosis) and spent a few years in a hospital. Some says that a very similiar motif appearing in "Tonari no Totoro" ("My neighbour Totoro") was strongly inspired by that part of his life. Also, during his childhood, Miyazaki had to change his place of residence and switch schools several times, which influenced his films a lot, too.
It was common in postwar Japan that children and teenagers wanted to become manga artists and Miyazaki wasn't an exception. He was interested especially in works of Osamu Tezuka and it took plenty of time until Miyazaki got rid of a great Tezuka influence in his creations.
In 1963 Miyazaki graduated from Gakushuin University (degrees in political science and economics) and in the same year he got a job at Toei Animation and gained his first recognition on "Gulliver's Travels Beyond the Moon", a Toei production, in 1965, as an in-between artist. Since 1968 he's started getting more important roles in animating, as a chief animator, concept artist, scene designer and storyboarder.
Studio Ghibli, an animation and film studio, was founded in 1985 by Miyazaki and Isao Takahata. Its logo features a forest spirit Totoro, from the movie "Tonari no Totoro", which is said to be the most famous Ghibli production. So far, the studio has won 3 Animage Anime Grand Prix Awards, a Golden Bear and an Oscar for Best Animated Feature (for "Spirited Away", which is the only film outside the English-speaking world that got this prize). Miyazaki himself was a director of 8 from 18 feature films from this studio (or 9/19 if you count "Nausicaä of the Valley of the Wind" in).
His dedication to the studio and to animation in general has been reported to effect negatively on his family relations. However, Hayao's son, Gorō, decided to follow his father's path and since then he's directed two films in Studio Ghibli, "Tales from Earthsea" (2006) and "From up on Poppy Hill" (2011). Gorō is supported by his father, despite the fact that Hayao said that "he doesn't want to create a dynasty of animators".
Miyazaki's films often include pacifist themes, a border and relationship between nature and technology. The main character is usually an independent girl or young woman. Despite the usual bright and positive general look, they hide much less positive and bright elements underneath - or, according to some other Ghibli movies, it's vice versa. Nevertheless, watching those films is always a great way to spend your time.