Kidd is best known for his movie work, but he was probably even more important and influential on stage. As you can see from the obituary, he is often linked with Jerome Robbins, in that they were both young, brilliant, innovative ballet choreographers who made an easy transition to Broadway choreography and eventually to directing. There had been other ballet people who did well on Broadway, like George Balanchine, Agnes De Mille and Kidd's mentor Eugene Loring. But they were limited to being choreographers; De Mille tried to break into directing but didn't really succeed. (She was credited as the director as well as the choreographer of Rodgers and Hammerstein's Allegro, but Hammerstein took over directing many of the book scenes himself.) Robbins and Kidd became directors and not only became fully successful and authoritative directors, they made a case for the advantages of having one person in charge of the whole production, including dancing.
You could argue that Kidd's Li'l Abner was the first show to really make a case for one man having full control over a musical. Robbins had already done Peter Pan with Mary Martin, but it wasn't really a hit at the time; and while Robbins directed Bells Are Ringing the same year as Kidd did Li'l Abner, Robbins split the dance duties with Bob Fosse. Li'l Abner was Kidd's show all the way: not only did he direct it and choreograph it, he was one of the producers. Kidd selected the composer and lyricist (bringing over the same people who had done the songs for Seven Brides For Seven Brothers, which he choreographed). Traditionally, the staging of a Broadway musical was a mix of at least two people's ideas; it had to be, because a regular stage director was equipped to approve the dances but not to create them. Li'l Abner, which would never have worked if the larger-than-life tone had let up for even a second, showed what could happen if one person was staging all the scenes himself; it had been done on film to some extent (like in the movies directed and choreographed by Gene Kelly and Stanley Donen) but not really on stage, and that paved the way for super-director productions like West Side Story, Gypsy, Fiddler On the Roof (Robbins) or Sweet Charity and Chicago (Bob Fosse), or even A Chorus Line (Michael Bennett) shows that were all part of one vision from beginning to end.
Kidd also had a big influence, on film and stage, on choreography: his famous athletic moves were copied all over the place. For example, the "Step In Time" sequence from Mary Poppins was choreographed by Kidd's former assistants, Marc Breaux and Dee Dee Wood (who re-created Kidd's choreography when Kidd didn't do the film of Li'l Abner), and is essentially imitation Kidd.
For some real Michael Kidd, here's the big number from the only movie he directed, Merry Andrew with Danny Kaye. The song is "Salud" by Johnny Mercer (Kidd's favorite lyricist, and for good reason) and Saul Chaplin (who had arranged the music for Seven Brides), led by Kaye and Salvatore Baccaloni; the dancers doing Kidd's trademark death-defying steps in his equally trademark color-co-ordinated costumes include Tommy Rall.
And from the original cast of Destry Rides Again (1959), directed and choreographed by Kidd: