Peck was born in the Hell's Kitchen area of New York City, New York. His parents were unmarried at the time of his birth and Peck never knew his father. He grew up with his mother, Barbara,who is a career coach. Josh had asthma during his childhood, and often stayed indoors watching old sitcoms. He has described himself as a "Jewish kid that grew up in New York City just trying to hustle and make it in life". He performed in children's musical theater at New York's TADA! Youth Theater when he was nine years old. He was inspired to become involved in stand-up comedy when he was eight years old.

Peck subsequently appeared in local children's theater, as well as performed stand-up comedy at the Caroline's Comedy Club for the Audrey Hepburnfoundation. In 1999, he was offered a role on Nickelodeon's The Amanda Show and, at his mother's suggestion, accepted the part and moved to San Francisco to further pursue an acting career.

Peck made his film debut in the 2000 film, Snow Day, and appeared regularly on The Amanda Show until the end of its run in 2002. He also starred opposite Alex D. Linz and Zena Grey in the theatrical film, Max Keeble's Big Move, which was released on October 5, 2001. In 2001, he guest starred in an episode of the popular NBC drama ER called Thy Will Be Done. During this time period, Peck appeared in several independent films, including Spun and 2004's Mean Creek, for which he received critical praise. He was cast as Josh Nichols, opposite Drake Bell's Drake Parker, in another Nickelodeon sitcom, Drake & Josh, which began airing in 2004 and gained Peck recognition among young audiences. In 2006, Bell and Peck starred in their own TV movie called Drake & Josh Go Hollywood, which was a huge success, and in 2007, they starred in a sequel called Drake & Josh: Really Big Shrimp. He was nominated for Favorite Television Actor at the 2008 Nickelodeon Kids' Choice Awards for his work on Drake and Josh. Peck also made his debut as a director in the Drake & Josh episode, Battle of Panthatar. Josh will continue to direct as he directs an episode of iCarly in its next season, according to Miranda Cosgrove Peck's co-star in Drake & Josh and star of iCarly. He has also appeared in the series finale of What's New, Scooby-Doo?, an episode of Codename: Kids Next Door called Operation: C.A.K.E.D.-F.I.V.E., and the direct-to-video New Line Cinema film Havoc.

In 2006, Peck appeared in the independent film Special, which premiered at the Sundance Film Festival, and voiced Eddie, one of two possum brothers, in the animated sequel, Ice Age: The Meltdown, which was released on March 31 of that year. Peck played a high school bully in 2008's Drillbit Taylor, and will star in the upcoming films The Wackness, American Primitive, Safety Glass, and Ice Age: Dawn of the Dinosaurs.