The Inca Empire effectively ended in 1533 with the death of Atahualpa. The Spanish executed the Inca ruler on 29 August 1533 after capturing him at Cajamarca and accepting a vast ransom of gold and silver.
Atahualpa’s death shattered the empire’s central authority at a moment when it was already weakened by a civil war between Atahualpa and his brother Huascar. Francisco Pizarro’s forces exploited that division, entered the imperial power structure, and installed Manco Inca Yupanqui as a nominal ruler.
The date needs a small qualification. Inca political resistance did not disappear immediately: Manco Inca later established the Neo-Inca State at Vilcabamba, which survived in the Andes for decades. Spanish forces conquered that last stronghold in 1572 and executed its ruler, Túpac Amaru. Thus 1533 marks the effective collapse of the main empire, while 1572 marks the end of the last independent Inca state.