Robbins was born Jerome Wilson Rabinowitz, exactly one month before the end of World War I, in the Jewish Maternity Hospital in the heart of Manhattan’s Lower East Side – a neighborhood populated by many immigrants. The Rabinowitz family lived in a large apartment house at 51 East 97th at the northeast corner of Madison Avenue. Known as "Jerry" to those close to him, Robbins was given a middle name that reflected his parents' patriotic enthusiasm for the then-president. Rabinowitz, however, translates to “son of a rabbi”, a name Robbins never liked, since it marked him as the son of an immigrant. So he took the name "Robbins".

In the early 1920s, the Rabinowitz family moved to Weehawken, New Jersey. 10 years earlier, Fred and Adele Astaire had lived there briefly as children, only a block away from one of Robbins's boyhood homes. His father and uncle opened the “Comfort Corset Company,” a unique venture for the family, which had many show business connections, including vaudeville performers and theater owners.

Robbins began college studying chemistry at New York University (NYU) but dropped out after a year for financial reasons and to pursue dance. He studied at the New Dance League, learning ballet with Ella Daganova, Antony Tudor and Eugene Loring; modern dance; Spanish dancing with the famed Helen Veola; folk dance with Yeichi Nimura; and dance composition with Bessie Schoenberg.

By 1939, Robbins was dancing in the chorus of such Broadway shows as Great Lady, The Straw Hat Revue, and Keep off the Grass, which George Balanchine choreographed. Robbins was also dancing and choreographing at Camp Tamiment in the Poconos of Pennsylvania. Here he choreographed many dramatic pieces with controversial ideas about race, lynching, and war. But in 1940, he turned his back (albeit temporarily) on the theater and joined Ballet Theatre (later known as American Ballet Theatre). From 1941 through 1944, Robbins was a soloist with the company, gaining notice for his Hermes in Helen of Troy, the Moor in Petrouchka and Benvolio in Romeo and Juliet.

At this time, Broadway dance was changing. Agnes de Mille had brought not just ballet to Oklahoma! but had also made dance an integral part of the drama of the musical. Challenged, Robbins choreographed and performed in Fancy Free, a ballet about sailors on liberty, at the Metropolitan Opera as part of the Ballet Theatre season in 1944. The inspiration for Fancy Free came from Paul Cadmus' 1934 painting The Fleet's In! which is part of the Sailor Trilogy. Robbins was recommended for a ballet based on the art work by his friend Mary Hunter Wolf. Distancing himself from the implicit homosexuality of that depiction, an element of controversy, Robbins said in an interview with The Christian Science Monitor, "After seeing...Fleet's In, which I inwardly rejected though it gave me the idea of doing the ballet, I watched sailors, and girls, too, all over town." He went on to say "I wanted to show that the boys in the service are healthy, vital boys: there is nothing sordid or morbid about them." Oliver Smith, set designer and collaborator on Fancy Free, knew Leonard Bernstein and eventually Robbins and Bernstein met to work on the music. This would be the first of several collaborative efforts. Fancy Free was a great success.

Later that year, Robbins conceived and choreographed On the Town (1944), a musical partly inspired by Fancy Free, which effectively launched his Broadway career. Once again, Bernstein wrote the music and Smith designed the sets. The book and lyrics were by a team that Robbins would work with again, Betty Comden and Adolph Green. His next musical was Billion Dollar Baby (1945). He was reportedly so unpopular by this point, that the company of this show watched silently as he backed up to the orchestra pit - and fell in. Two years later, he received plaudits for his hilarious Keystone Kops ballet in High Button Shoes (1947), including his first Tony Award for choreography.

During this period, Robbins continued to create dances for the Ballet Theatre, alternating between musicals and ballet for the better part of the next two decades. Barely a year went by without a new Robbins ballet and a new Robbins musical. With George Balanchine, he choreographed Jones Beach at the City Center Theater in 1950, and directed and choreographed Irving Berlin's Call Me Madam, starring Ethel Merman.

In 1951, Robbins created the now-celebrated dance sequences in Rodgers and Hammerstein's The King & I (including the March of the Siamese Children, the ballet The Small House of Uncle Thomas and the "Shall We Dance?" polka between the two leads). That same year, he created The Cage for the New York City Ballet, with which he was now associated. He also performed, uncredited, show doctoring on the musicals A Tree Grows in Brooklyn (1951), Wish You Were Here (1952), and Wonderful Town (1953).

Robbins collaborated with George Abbott on The Pajama Game (1954), which launched the career of Shirley MacLaine, worked on the 1955 Mary Martin vehicle, Peter Pan (recreated for the small screen in 1955, 1956 and 1960) and directed and co-choreographed (with Bob Fosse) Bells Are Ringing (1956), starring Judy Holliday. In 1957, he conceived, choreographed, and directed a show that some feel is his crowning achievement: West Side Story.

West Side Story is a contemporary version of Romeo and Juliet, set in Hell's Kitchen. The show, with music by Leonard Bernstein, marked the first collaboration between Robbins and Stephen Sondheim, who wrote the lyrics, as well as Arthur Laurents, who wrote the book. To help the young cast grow into their roles, Robbins did not allow those playing members of opposite gangs (Jets and Sharks) to mix during the rehearsal process. The original Broadway production featured Carol Lawrence as Maria, Larry Kert as Tony and Chita Rivera as Anita. Although it opened to good reviews, it was overshadowed by Meredith Willson's The Music Man at that year's Tony Awards. West Side Story did, however, earn Robbins his second Tony Award for choreography, and is now hailed as a groundbreaking classic.