Short answers, pulled from the story.
Jacob Burckhardt is best known for The Civilization of the Renaissance in Italy, published in 1860. It was the most influential interpretation of the Italian Renaissance in the nineteenth century and is still widely read. He is also recognized as one of the founding figures of cultural history as an academic discipline.
Burckhardt taught at the University of Basel from 1843 to 1855, then at the Federal Polytechnic School, and returned to Basel in 1858 to hold a professorship until his retirement in 1893. He twice declined offers from German universities, turning down chairs at Tübingen in 1867 and at the University of Berlin in 1872.
Nietzsche was appointed professor of classical philology at Basel in 1869 at the age of twenty-four and attended some of Burckhardt's lectures. Both men admired Arthur Schopenhauer, and their extensive correspondence has been published. Burckhardt enjoyed Nietzsche's company while keeping a deliberate distance from his evolving philosophy.
Burckhardt predicted that violent demagogues, whom he called "terrible simplifiers," would play central roles in a cataclysmic twentieth century. He made this prediction well before that century arrived. He also wrote extensively about the dangers of growing European nationalism and militarism in his lectures and writings.
Der Cicerone, published in 1855 and dedicated to the art historian Franz Theodor Kugler, was a guide to sculpture, architecture, and painting in Italy. It was described as "the finest travel guide that has ever been written" and became an indispensable reference for art travellers. About half of the original edition was devoted to the art of the Renaissance.
Burckhardt treated art, literature, and architecture as primary sources for understanding historical periods, rather than limiting history to politics and records of state. His approach was deliberately unsystematic and opposed to Hegelianism, economism, and positivism. Sigfried Giedion credited him as the first to show how a period should be treated in its entirety, including the social institutions of daily life.