The New York School of photography is not particularly related to the more well-known New York School of abstract expressionism. Although there's some overlap in time (and of course place), the photographic movement began earlier and has different characteristics.

The Michael Hoppen Gallery in London had a New York School exhibit in 2008, and from their description:

Between the late 1930s and the early 1960s a group of young photographers living and working in New York City redefined street photography. This group of artists became known as The New York School.

These photographers documented the post war energy and exotic chaos of New York City as it evolved from the crisis years of the Great Depression and the Second World War through to the social turbulence of the early seventies. Most of them worked on magazines but it was their personal work that stood them apart. They captured the choreography of the city from the sidewalks of downtown, to the intensity of Times Square, the isolation and elegance of the architecture and the mass of humanity at Coney Island. Many of the New York School identified with the values of film noir, stylish low-key black and white images with a certain moral ambiguity. Their style utlilised the methods of documentary journalism, small cameras, available light and a sense of the fleeting and candid and yet they rejected the anecdotal descriptiveness of most photojournalism.

There was also an exhibit titled On the Street: The New York School of Photographers in 2004-2005 at the Phoenix Art Museum in Arizona, and an article on that exhibit reads in part:

New York — its streets, attractions, people, energy and urban charm — is the focus of a new exhibition organized by the Center for Creative Photography. [...] Street: The New York School of Photographers presents the work of such renowned as artists as Diane Arbus, Roy deCarava, Robert Frank, Lee Friedlander, William Klein, Helen Levitt, Lisette Model, Weegee, Garry Winogrand and others who lived and worked in New York in the twentieth century. They documented the drama, turbulence, exoticism and humanity of this great metropolis as it evolved in the crucial years from the economic crisis of the Depression through the social unrest of the early seventies. [...]

Generally defiant and not given to "isms," artists of the New York School betray an existential attitude in their work. Photographers such as the famous Weegee clearly put social realism -- an effort to describe the city in blunt, human terms -- at the top of their agendas. [...] [Earlier] artists adduce in their work the promise of the city, while the photographers later dubbed the " New York School " more often show us the fallout, that is, the shortcomings of this promise.

Additionally, the book The New York School: Photographs, 1936-1963, by Jane Livingston, collects a number of photographs by sixteen photographers associated with the school, and the publisher's copy reads:

The New York School of Photography refers to a loosely defined group of photographers who lived and worked in New York City during the 1930s, 1940s and 1950s sharing influences, subjects and stylistic earmarks. [...] These photographers, many of whom also worked for the magazines of the day, stretched the boundaries of their medium in their personal work as street photographers. As their subjects, they chose the random choreography of New York's sidewalks, the crush of bodies on Coney Island's beaches and the glare of festival lights and neon signs. Many of the photographers identified with the look and values embodied in "film noir".

So, in general, this group spanned a 30-year period roughly from the 1930s to the 1960s. It's a movement of street photography, tending towards a stylized but very gritty depiction of reality. In contrast to straight photojournalism, images from the school often have an air of ambiguity and mystery. Of course, black and white film seems to be universal, fitting with the film noir aesthetic, and perhaps it's not coincidental that the end of the school coincides with the rise of color film in art-world photography, as exemplified by William Eggleston.

From the three sources above, photographers in the movement include:

• Diane Arbus
• Richard Avedon
• Alexey Brodovitch
• Ted Croner
• Bruce Davidson
• Roy deCarava
• Don Donaghy
• Louis Faurer
• Lee Friedlander
• Robert Frank
• Sid Grossman
• William Klein
• Saul Leiter
• Leon Levinstein
• Helen Levitt
• Neil Libbert
• Lisette Model
• Louis Stettner
• David Vestal
• Weegee
• Garry Winogrand

That's a lot of important names, and clearly they had a huge influence on photography (and particularly on street photography). However, if you read most biographies of any one of them, you'll not always find references to "the New York School" — it seems that the name and grouping is more a retroactive classifier used by some people studying and collecting photography than a self-identified movement (in fact, it may have been coined by the writer of the book referenced above). However, there is significant overlap with the Photo League, which may be thought of as a precursor, with the New York School less political but more a part of the art world.