Video report by ITV News Correspondent Dan Rivers
A bystander filmed the arrest of the man - identified as George Floyd by a lawyer representing his family - which showed Floyd pleading that he could not breathe as a white officer knelt on his neck.
The four police officers involved in the arrest have been fired, a move Minneapolis Mayor Jacob Frey said was "the right call."
Floyd's death on Monday night is under investigation by the FBI and state law enforcement authorities.
Hundreds of protesters gathered near the site of the arrest on Tuesday, holding "black lives matter" placards and signs calling to "end police brutality".
Minneapolis Mayor Jacob Frey apologised to the black community on Tuesday in a post on his Facebook page.
He wrote: "Being Black in America should not be a death sentence.
"For five minutes, we watched a white officer press his knee into a black man’s neck. Five minutes. When you hear someone calling for help, you're supposed to help. This officer failed in the most basic, human sense".
Police said Floyd matched the description of a suspect in a forgery case at a grocery store, and that he resisted arrest.
In the video shared widely online, the unidentified officer is seen ignoring his pleas as Floyd is heard saying: "Please, please, please, I can’t breathe. Please, man".
After several minutes, one of the officers tells Floyd to "relax".
Minutes pass, and Floyd becomes motionless under the officer's restraint - the officer leaves his knee on his neck for several minutes more.
The man who died was named as George Floyd by Ben Crump, a prominent civil rights and personal injury lawyer, who has been hired by Mr Floyd's family.
In a statement, Mr Crump said: "We all watched the horrific death of George Floyd on video as witnesses begged the police officer to take him into the police car and get off his neck.
"This abusive, excessive and inhumane use of force cost the life of a man who was being detained by the police for questioning about a non-violent charge."
The video shows several witnesses gathered on a nearby pavement - some recording the scene on their phones.
The bystanders become increasingly agitated as the man pleads with police. One bystander told officers they need to let him breathe. Another yelled at them to check the man's pulse.
Minneapolis Police Chief Medaria Arradondo said the department would conduct a full internal investigation.
Chief Arradondo told reporters: "What occurred last night is certainly very tragic, and very sad.
"The vast majority of the work we do does never require the use of force. So, again, my condolences, deepest condolences to him and his family members."
Leslie Redmond, Minneapolis NAACP President said: "Mr. Floyd should not have died. He should still be alive today. And we urge you all to stand up.
"This is not a black people issue, this is a human issue. And if you are a human being, you should be outraged about what happened yesterday."
In Minneapolis, kneeling on a suspect's neck is allowed under the department's use-of-force policy for officers who have received training in how to compress a neck without applying direct pressure to the airway.
It is considered a "non-deadly force option," according to the department's policy handbook.