Police identified the man as Akram Joudeh, 32, whose most recent address was in Queens, New York.
When police officers approached Joudeh -- around 5 p.m. -- he pulled out the meat cleaver and ran away. The officers chased Joudeh, and shot him with a Taser, but that did nothing to immobilize him. They shouted at him to drop the meat cleaver.
As Joudeh ran down the street, an off-duty police detective attempted to tackle him. Joudeh reacted by slashing the detective's face, police said.
The detective is listed in serious condition, James O'Neill, the NYPD's top chief, said Thursday night. The detective was being treated for a six-inch gash from his temple to his jaw.
Police then opened fire. Police fired 18 shots, striking Joudeh at least twice. "They shot until the threat was stopped," said O'Neill.
Joudeh was taken to Bellevue Hospital on Manhattan's East Side, where he was listed in critical but stable condition. Police said Joudeh has more than a dozen prior arrests.
Two additional officers sustained non-life threatening injuries during the encounter. It has not been determined how these injuries were sustained, O'Neill said.
ABC News' Darren Reynolds contributed to this report.