The "New Look" Batman was new Batman editor Julius Schwartz's effort to revitalize and modernize the Batman comics for the 1960s. In the space of one issue, Batwoman, Bat-Mite, and the rest of the Batman family introduced in the late 1950s were gone from the books. The stories shifted from science fiction to those that emphasized Batman's detective skills. The art of the stories was also modernized, with Carmine Infantino on Detective Comics and veteran Batman artist Sheldon Moldoff's pencils on Batman slicked up by Joe Giella. You can see clearly see the shift by comparing the covers of the Batman comics cover dated May 1964 with the covers of the Batman comics cover dated June 1964.
I tend to think of it as Batman having two Silver Ages. There's the Silver Age Batman from the late fifties to early sixties with the science fiction tone and expanded Batman family and then there's the Silver Age Batman from the late sixties with the "New Look" revamp and the stories that reflect the camp tone of the 1960s television show. So if I blog about a Batman story from the sixties, it will be one from the first Silver Age Batman had and not from his second, "New Look" Silver Age.