The explosion occurred in Bishop Street at about 20:15 GMT on Saturday, shortly after police received a warning.
Officers were on the scene when the blast occurred. No one is believed to have been injured.
Police had just started evacuating nearby buildings, including a hotel, when the explosion took place. A cordon has been set up at the scene.
Local politicians have told the BBC that the vehicle was hijacked in Derry some time before the explosion.
An eyewitness told the BBC: "The remains of the car could still be seen burning in the middle of the road.
"A cordon has been set up around the scene and police are evacuating more buildings on Bishop Street."
In a statement issued on Facebook, the PSNI said: "As far as we know no one [is] injured."
However, they said that there is another car they "are not happy about" and are evacuating the area and advising residents in the area to make preparations to leave.
Local resident Greg McLaughlin said his windows shook with the force of the blast.
"It was very, very loud. I knew right away this was a bomb," he said.
"We knew it was quite close.
"You could see the ball of fire on the street. It sounded to me like a very significant blast. I haven't heard anything like it in Derry for quite a while."
At the scene, BBC reporter Dean McLaughlin
Businesses and other properties have been evacuated and the cordon on Bishop Street has been extended.
One business owner said she heard a bang and was lucky to be inside at the time. She said she could have been on the street minutes later.
Bishop Street is within Derry's famous city walls.
The PSNI has asked members of the public to stay away.
Colin Neill, chief
executive of Hospitality Ulster, said the explosion had caused major disruption in the Bishop Street area of the city.
"This has to be condemned in its strongest terms, taking place on a Saturday evening when many people are out enjoying themselves.
"The loss of trade for the local hospitality sector and the impact that this will have on tourism and the local economy benefits no one."
The Northern Ireland secretary, Karen Bradley, said she was concerned at the reports coming from Londonderry and was "being kept informed by PSNI".
Sinn Féin MP Elisha McCallion said the incident had "shocked the local community".
"In particular, there are many elderly residents who live in the area who have been alarmed by this incident," she said.
SDLP MLA Mark H Durkan tweeted: "Whoever is responsible for this explosion outside Bishop Street Courthouse in Derry clearly hasn't got the message that the people of Derry DO NOT want this on our streets."
The DUP MLA, Gary Middleton, described the incident as a "disgraceful act of terrorism".
His party leader Arlene Foster tweeted: "This pointless act of terror must be condemned in the strongest terms. Only hurts the people of the city.
"Perpetrated by people with no regard for life. Grateful to our emergency services for their swift actions which helped ensure there have been no fatalities or injuries."
The Irish tánaiste (deputy prime minister) Simon Coveney condemned the attack.
"There is no place and no justification possible for such acts of terror, which seek to drag Northern Ireland back to violence and conflict," he wrote on Twitter.
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