The Prime Minister said: "A great light has gone out in the world. Nelson Mandela was a towering figure in our time: a legend in life and now in death - a true global hero.
"I believe that his inspiration for the future will be every bit as powerful as the extraordinary things he achieved in his remarkable life."
MPs will be given the opportunity to pay their tributes to Mr Mandela in the House of Commons on Monday.
US President Barack Obama said he had achieved "more than can be expected by any man".
"Today he has gone home and we have lost one of the most inspirational, courageous and profoundly good human beings that any of us will share time with on this Earth," he said.
"He now no longer belongs to us, he belongs to the ages."
South African President Jacob Zuma made the emotional announcement live on television, in which he said Mr Mandela was now at peace.
He added: "Our nation has lost its greatest son."
FW de Klerk, South Africa's last white president, who shared the Nobel Peace Prize with Mr Mandela in 1993, said he was a "great unifier".
He added: "This emphasis on reconciliation was his greatest legacy."
South Africa's archbishop emeritus Desmond Tutu said South Africa was "drowned by grief".
He said: "He was a unifier from the moment he walked out of prison."
French President Francois Hollande added: "Nelson Mandela's message will not disappear. It will continue to inspire fighters for freedom, and to give confidence to peoples in the defence of just causes and universal rights."
Australian Prime Minister Tony Abbott said: "He spent much of his life standing against the injustice of apartheid.
"When that fight was won, he inspired us again by his capacity to forgive and reconcile his country.
"While the world may never see another Nelson Mandela, he has inspired countless men and women throughout the world to live more courageous and honest lives."
Scotland's First Minister Alex Salmond described him as a "towering statesman" whose influence "transcended ideology, race and creed".
Irish Taoiseach Enda Kenny said a "great light has been extinguished".
He said: "The boy from the Transkei has finished his long walk. His journey transformed not just South Africa, but humanity itself."
UN Secretary General Ban Ki Moon said: "Only because of such a great man like Nelson Mandela is it possible that particular people in Africa and elsewhere are able to enjoy freedom and human dignity.
"We have to learn the wisdom and determinations and commitment of Mr Mandela to make this world better for all."
Prince William and the Duchess of Cambridge also paid tribute at the Royal premiere of the new Nelson Mandela film, Long Walk To Freedom, in Leicester Square, London.
The Duke of Cambridge said: "It is extremely sad and tragic news. We are just reminded of what an extraordinary and inspiring man Nelson Mandela was."
Leading figures from around the world have also been joining in with praise for the anti-apartheid icon.
Former US president George W Bush said: "He bore his burdens with dignity and grace, and our world is better off because of his example."
Former prime minister Tony Blair described him as the world's "most powerful symbol of reconciliation, hope and progress".
Former US president Bill Clinton called Mr Mandela a "true friend".
He said: "History will remember Nelson Mandela as a champion for human dignity and freedom, for peace and reconciliation."
US Secretary of State John Kerry said: "Now that his long walk has ended, the example he set for all humanity lives on. He will be remembered as a pioneer for peace."
Labour Party leader Ed Miliband said: "Above all, he showed us the power of people, in the cause of justice, to overcome the mightiest obstacles. He moved the world and the world will miss him deeply."
Deputy Prime Minister Nick Clegg said: "The hope he offered was enough to unite races; it bridged cultures and transcended generations; and it could heal the deepest divides."
Foreign Secretary William Hague said: "An international icon and inspiration to millions, his appeal transcended race, religion and class. His courage, humility and sense of forgiveness have secured his place in history."
London Mayor Boris Johnson said: "When the definitive history of our time is written, the name Mandela will stand taller than most - perhaps tallest of them all."
A joint message from President of the European Commission, Jose Manuel Barroso, and President of the European Council, Herman Van Rompuy, said: "We mourn the death of one of the greatest political figures of our times.
"Nelson Mandela represents the fight against racism, political violence and intolerance. Only a person with his profound humanity, moral integrity and authority and clear vision for the future of his country, could have achieved this."
American civil rights leader Jesse Jackson said Mr Mandela has left an "everlasting imprint".
He said: "Nelson Mandela was a giant of immense and unwavering intellect, courage and moral authority. He chose reconciliation over retaliation. He challenged the course of history."
Israeli Prime Minister Benjamin Netanyahu said he was the "father of his people", adding: "He will be remembered as the father of new South Africa and as an outstanding moral leader."
German Chancellor Angela Merkel said: "Nelson Mandela's shining example and his political legacy of non-violence and the condemnation of all forms of racism will continue to inspire people around the world for many years to come.
Mexican President Enrique Pena Nieto said: "Humanity has lost a tireless champion of peace, liberty and equality."
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