Short answers, pulled from the story.
Lorenzo di Piero de' Medici (the 12th of September 1492 - the 4th of May 1519) was the ruler of Florence from 1516 until his death and Duke of Urbino during the same period. He was the grandson of Lorenzo the Magnificent and the father of Catherine de' Medici, who became Queen of France.
Lorenzo persuaded his uncle Pope Leo X to grant him the Duchy of Urbino in 1516, when he was twenty-four years old. This triggered the War of Urbino against the previous duke, Francesco Maria I della Rovere, which Lorenzo eventually settled by treaty in September 1517.
Catherine de' Medici's father was Lorenzo di Piero de' Medici, Duke of Urbino. Her mother, Madeleine de La Tour, died on the 28th of April 1519, and her father died six days later, leaving Catherine to be raised by the Medici popes Leo X and Clement VII.
Niccolò Machiavelli dedicated The Prince to Lorenzo to advise him on tactics to maintain his authority as ruler of Florence and Duke of Urbino. The dedication made Lorenzo the named recipient of one of the most influential works of political writing in European history.
Lorenzo II is buried in the Medici Chapel of Florence's Church of San Lorenzo. There is disagreement among historians over which of the two tombs is his, with scholar Richard Trexler arguing that the traditional identification of the figures in Michelangelo's sculptures may be reversed.
Alessandro de' Medici (1510-1537) was the recognised but illegitimate son of Lorenzo Duke of Urbino, born to a servant girl named Simonetta. After Lorenzo's death, Alessandro inherited his estate and became the first Duke of Florence before being assassinated by his cousin Lorenzaccio in 1537.