Short answers, pulled from the story.
Lorenzo di Pierfrancesco de' Medici (the 4th of August 1463 - the 20th of May 1503) was an Italian banker and politician from the junior, or Popolani, branch of the Medici family of Florence. He was a prominent patron of Renaissance art, a student of Marsilio Ficino and Angelo Poliziano, and a figure in the republican government of Florence after 1494.
Amerigo Vespucci was a fellow student of Lorenzo di Pierfrancesco under the humanist Giorgio Antonio Vespucci, Amerigo's uncle. From 1483, Amerigo became an employee and personal friend of Lorenzo di Pierfrancesco. In the early 1500s, Amerigo Vespucci sent most of his famous letters about the "New World" to Lorenzo di Pierfrancesco.
Lorenzo di Pierfrancesco de' Medici is believed to have commissioned Botticelli's The Birth of Venus. He also commissioned an illuminated manuscript of Dante's Divine Comedy featuring Botticelli's artwork around 1485, and Botticelli's Allegory of Spring is associated with his 1482 wedding to Semiramide d'Appiani.
Lorenzo di Pierfrancesco and his brother Giovanni received the nickname Popolano, meaning "of the people", because they allied with the Republican faction in Florence after 1494. They supported the republican government that expelled Piero de' Medici following King Charles VIII of France's invasion of Italy.
After the death of Lorenzo di Pierfrancesco's father in 1476, Lorenzo il Magnifico became the guardian of the Popolani boys. Facing financial difficulties after 1478, il Magnifico used "forced loans" from the brothers' inheritance. By the time they came of age, they claimed he owed them over 100,000 ducats; il Magnifico eventually repaid a little over half that amount in 1485.
Lorenzo di Pierfrancesco's grandson Lorenzino de' Medici assassinated Alessandro de' Medici, the last ruler of Florence from the senior Medici branch. This transferred power to Lorenzo di Pierfrancesco's great-grandson Cosimo I de' Medici, who became the dominant ruler of Tuscany.