I just came back from Invictus. Invictus is about the South African rugby team, which Nelson Mandela saw as a means to unite his nation. At the beginning of his presidency, black South Africans viewed the team as a symbol of Apartheid, so they mostly rooted for its opponents. Nelson Mandela himself rooted for its opponents for many years, but he changed his mind once he became President.
His aim was to reconcile whites and blacks. Part of his reason was practical, for he recognized that the white minority still owned the police and much of the economy, so it needed to be appeased. But he also sincerely believed in the power of forgiveness. To his black security guard, who didn't like the idea of working alongside white security agents, Mandela said that forgiveness can free a person from fear. To his daughter, who remarked that the Matt Daimon character looked like the white policemen who threw the Mandelas out of their home, Nelson said that he had to put the well-being of the nation ahead of personal resentment.
Even in his day-to-day life, Nelson Mandela was humble and loving to those around him. When Matt Daimon asked a white security guard what kind of man Mandela is, the guard replied that the previous president didn't know who he was, but Mandela brought him his favorite food from England.
There are many powerful moments in the movie: the black child dancing with the white police officers after the rugby team's victory, Matt Daimon visiting Nelson Mandela's prison cell (which, according to Roger Ebert, actually was Mandela's prison cell), etc. But the scenes that had the most impact on me were the ones in which Mandela walked out into the stadium with a wide smile, hunched over because of his old age, waving as some booed him. That humanized Mandela for me, so I couldn't help but like him.
I didn't know anything about Rugby before I saw this movie, but it looks like football, only without the protective gear. Ouch!