"His commitment to a democratic, multi-racial South Africa, his steadfast pursuit of justice, his willingness to reconcile with those who persecuted him most - these are just some of the hallmarks of a remarkable man," Ban said.
"May we all continue to benefit from his wisdom, his good works and good humour for many years to come."
Mandela stepped down after serving one term as president in South Africa, the first black South African to hold the post.
Since 1999, he has devoted himself to causes such as fighting Aids and poverty and championing the rights of children and in recent years has stressed that if his legacy is to live, others must take up his causes.
Across South Africa, people collected clothing for poor children, painted schools, planted trees near Mandela's boyhood home and renovated a building in downtown Johannesburg for people left homeless by a fire.
Events for reading to the blind, distributing blankets to the homeless and refurbishing homes for Aids orphans were also planned.
Mandela Day organisers encouraged people to devote at least a minute for each of the 67 years Mandela campaigned against apartheid to community service.
Thandiwe Gwinza, a nurse in Johannesburg, went straight from night shift at a hospital to volunteer at a soup kitchen.
"This morning what I have done for 67 minutes, I have helped the community making soup," she said.
"I have made soup for the people and I have given them my old clothes."
Jacob Zuma, South Africa's president and the leader of the ruling African National Congress, paid a birthday visit to Mandela at his home in Johannesburg and then went to a poor neighbourhood in the city to visit elderly South Africans at a Mandela Day lunch.
The opposition Democratic Alliance also got involved, with Helen Zille, the party's leader, serving at a soup kitchen in Cape Town.
At a Mandela Day concert in New York, in the US, acts including Stevie Wonder, Alicia Keys and Aretha Franklin are to perform for the benefit of Mandela's Aids foundation.