What particular event took place that made Nelson Mandela want things to be different?

One event that motivated Nelson Mandela to turn to more radical means of changing South Africa during Apartheid was the Sharpeville Massacre of 1960. At this point, Mandela was already a long-time member of the ANC, or African National Congress, formed to fight against Apartheid, and he had been arrested in 1956 and tried (but acquitted) on the charge of treason.
In the Sharpeville Massacre, the police fired on protestors who were peacefully demonstrating against pass laws, and 69 people were killed. In the aftermath of the event, the country was convulsed by riots and unrest. The Sharpeville Massacre had motivated him to turn to new means of creating change. In 1961, Mandela became the founder and leader of Umkhonto we Sizwe (Spear of the Nation), a paramilitary wing of the ANC. In 1962, Mandela was jailed for 5 years for instigating a workers' strike, and the following year, he was arrested in the so-called Rivonia Trial for threatening to overthrow the government. Sentenced to life in prison in 1964, he served 27 years before his release. He went on to serve as President of South Africa. 

One of the primary events that occurred that made Nelson Mandela want things to be different was the white only election in 1948, in which the National Party won on the platform of apartheid. Another important moment came in 1950, with the violent government reaction to the joint strike called by the ANC, the Communist Party, and the South African Indian Congress to protest the ban of the Communist Party.
By the time of the election, Mandela was already involved with the African National Congress (ANC).  After the election, Mandela (along with other leaders such as Oliver Tambo and Walter Sisulu) began to lobby the ANC to protest militarily against the multitude of new laws enacted by the Nationalists.
In the bloody aftermath of the government crackdown against the so-called May Day strike, Mandela made the decision that freedom could only occur with a broad based alliance against apartheid.  The Africanist wing of the ANC disagreed, but Mandela held his ground and helped form the Congressional Alliance.
In 1952, the Congressional Alliance launched its Defiance Campaign.  While the campaign did not stop apartheid, it did solidify the ANC as a formidable force against white minority rule.
Source: South African History Online