In a televised address, Mr Zuma said: "Our nation has lost its greatest son. Our people have lost a father.
"What made Nelson Mandela great was precisely what made him human. We saw in him what we seek in ourselves."
Mr Zuma said: "Fellow South Africans, our beloved Nelson Rolihlahla Mandela, the founding president of our democratic nation has departed.
"He is now resting. He is now at peace.
"Our nation has lost its greatest son. Our people have lost a father."
Mr Zuma said all of South Africa's thoughts were with Mr Mandela's family, friends and those who fought alongside him during his struggle for equality.
"Our thoughts are with the South African people who today mourn the loss of the one person who more than any other came to embody their sense of a common nation," he said from Pretoria.
"Our thoughts are with the millions of people across the world who embraced Madiba as their own and who saw his cause as their cause.
"This is the moment of our deepest sorrow. Our nation has lost his greatest son. Yet what made Nelson Mandela great was precisely what made him human - we saw in him what we seek in ourselves and in him we saw so much of ourselves."
Mr Zuma said the man known as Madiba brought South Africa together and would be afforded a state funeral. South African flags would be lowered to half mast from tomorrow until the funeral, he said.
"As we gather to pay our last respects, let us conduct ourselves with the dignity and respect that Madiba personified," he said.
"Let us be mindful of his wishes and the wishes of his family as we gather wherever we are in the country and wherever we are in the world.
"Let us recall the values for which Madiba fought.
"Let us reaffirm his vision of a society in which none is exploited, oppressed or dispossessed by another."
Mr Zuma finished his address by saying simply: "We will always love you Madiba. May your soul rest in peace. God bless Africa."
Mr Mandela was the country's first black president and is known throughout the world for his anti-apartheid stance and led the nation's transition from white-minority rule in the 1990s, after 27 years in prison.
The statesman had been receiving medical care for a lung infection at his home, where hundreds of reporters tonight gathered ahead of rumours of his ill-health and where he was staying following a three-month hospital stay.
Since his hospital release, the South African presidency described Mr Mandela's condition as critical but stable, although rumours were this week circulated that he was on his death bed.
He was awarded the Nobel Peace Prize in 1993 and was elected South Africa's first black president in 1994.
David Cameron tweeted his condolences, adding: "A great light has gone out in the world. Nelson Mandela was a hero of our time. I've asked for the flag at No10 to be flown at half mast."
Mandela's extraordinary achievements during his lifetime meant he was never short of accolades from his peers.
:: "Nelson Mandela is a leader no prison cell, no intimidation, no threat could silence. A man whose belief in the future was so powerful that not even 27 years behind bars and barbed wire could destroy his dream that millions could be free" - Gordon Brown, former prime minister.
:: "Your sacrifices, your accomplishments and your many gifts to our world have affected those no longer with us, as well as those not yet born" - Former US president Bill Clinton and his wife, Secretary of State Hillary Clinton.
:: "How blessed we have been. He has become the most admired statesman in the world, an icon of forgiveness and reconciliation, a moral colossus ... wonderful man, wonderful, wonderful gift to us" - Archbishop Desmond Tutu.
:: "To this day, Madiba (Mandela) remains probably the single most admired, most respected, international figure in the entire world. He continues to inspire millions of people and several generations throughout the globe, by continuing to fight for reconciliation before recrimination, healing before bitterness, peace before conflict; by fighting for health, for education, for the right of every child to have a better start in life" - Former United Nations secretary general Kofi Annan.
:: "What is so special is that you spend 27 years in prison, you come out and you do the thing that everyone thought was impossible to do, become president of the nation and change the way people feel about Africa, and then you get to live to celebrate 90. I just don't know anybody who deserves this more or anybody who we should be gathered around literally at his feet to say thank you for all that he has given the world" - US chat show host Oprah Winfrey.
:: "He is a giant amongst the political figures of the modern age. At a time when we are looking for role models, what better role model than Nelson Mandela?" - Former Liberal Democrat leader Sir Menzies Campbell.
:: "Your name will forever be associated with the battle for justice, freedom and democracy. Your continuing fight to defeat HIV/Aids ensures you remain a beacon of hope for millions. Long after you deserved a break the struggle is still your life" - Former prime minister Tony Blair.
:: "He's someone who I often think about when things get tough and life isn't full of roses. Nothing can be as hard as what he went through" - Model Naomi Campbell.
:: "He has fought bravely ... and forgiven absolutely" - Formula One star Lewis Hamilton.
:: "I applaud the leadership he is providing on many vital causes in South Africa and around the world. Mr Mandela is a great example of courage, hope and the power of freedom" - Former US president George W Bush.