He had to apologise in 2005 for wearing a swastika armband to a party, which offended many Jewish people.
The video obtained by the News of the World shows Harry while still an officer cadet at Sandhurst military academy.
It was filmed in front of other cadets at an airport departure lounge as they waited for a flight to Cyprus to go on manoeuvres.
The newspaper said the prince, who is third in line to the throne, had called the soldier "our little Paki friend".
BBC royal correspondent Daniela Relph said this was an extremely embarrassing episode for the prince and the Royal Family.
She said the emergence of the three-year-old video was "unfortunate timing" for Harry, whose image had greatly improved since he served in Afghanistan last year.
Prince Harry and Ahmed Raza Khan, bottom, at Sandhurst in 2006
"That was a real step up for him, a real sense of maturity that people could see," she said.
Politicians and Muslim groups are among those to have condemned the prince's remarks.
Liberal Democrat leader Nick Clegg said the words would have caused "considerable offence", while Tory leader David Cameron said it was "a completely unacceptable thing to say".
Mohammed Shafiq, director of Muslim youth organisation the Ramadhan Foundation said: "I am deeply shocked and saddened at Prince Harry's racism. It has no justification.
"Prince Harry as a public figure must ensure that he promotes equality and tolerance and this rant whether today or three years ago is sickening and he should be thoroughly ashamed of himself."
Graham Smith from Republic, the campaign for an elected head of state said: "Harry Wales has not only demonstrated how he is unfit to be a possible future Head of State, he has shown he isn't even fit to be a leader in the armed forces."
"It is high time Harry was stripped of his title and privileges and withdrew from public life."
BBC Royal Correspondent Peter Hunt describes the content of the video
A statement from St James's Palace, with regards the term "Paki", said: "Prince Harry fully understands how offensive this term can be, and is extremely sorry for any offence his words might cause.
"However, on this occasion three years ago, Prince Harry used the term without any malice and as a nickname about a highly popular member of his platoon.
"There is no question that Prince Harry was in any way seeking to insult his friend."
The statement continued: "Prince Harry used the term 'raghead' to mean Taleban or Iraqi insurgent."
A man told BBC Radio Five Live the cadet concerned was his nephew, Ahmed Raza Khan, from Pakistan, who served with Prince Harry at Sandhurst for one year as a Commonwealth cadet.
Iftikhar Raja said his nephew, now a captain in the Pakistani army, would have risen above such terms and had not mentioned the incident to his family.
Mr Raja said: "At no time he told us that he was called Paki or he was a good friend of Prince Harry, I mean, although they served together that is true.
Iftikhar Raja: 'They should have had more respect for each other'
"But I myself am a British subject, I am proud to be British and if someone called me Pakistani I would be proud to be called that, but Paki is definitely a derogatory remark."
He added: "We expect better from our Royal Family on whom we spend millions and millions of pounds for training and schooling."
Captain Ahmed Raza Khan graduated with Harry from Sandhurst in 2006 receiving a special award from the Queen for being the best overseas officer cadet.
The Army has been trying to recruit soldiers from ethnic minority backgrounds as these are currently under-represented in the services.
A spokeswoman for the Ministry of Defence said: "Neither the Army nor the Armed Forces tolerates inappropriate behaviour in any shape or form.
"The Army takes all allegations of inappropriate behaviour very seriously and all substantive allegations are investigated.
"We are not aware of any complaint having been made by the individual. Bullying and racism are not endemic in the Armed Forces."
A spokeswoman for the Equality and Human Rights Commission said: "These appear to be disturbing allegations and we will be asking the MoD to see the evidence, share that evidence with us and their plans for dealing with it.
"We will then consider what further action might be necessary."
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