Short answers, pulled from the story.
Jacob Bernoulli is best known for deriving the first version of the law of large numbers, published posthumously in his work Ars Conjectandi in 1713. He also discovered the mathematical constant e, co-founded the calculus of variations with his brother Johann, and introduced the term "integral" with its modern meaning in a 1690 paper.
Ars Conjectandi is Jacob Bernoulli's most important work, a foundational text in probability theory. It was published in Basel in 1713, eight years after Bernoulli's death on the 16th of August 1705, and was incomplete at the time he died. The book introduced Bernoulli numbers and the concept of the Bernoulli trial, and derived the law of large numbers.
Bernoulli discovered the constant e in 1683 while studying compound interest. He observed that as the number of compounding intervals in a year grows larger, the value of a one-dollar account paying one hundred percent annual interest converges toward a fixed limit, which Leonhard Euler later named e and calculated as approximately 2.7182818.
Jacob and Johann Bernoulli initially collaborated, with Jacob tutoring his younger brother in mathematics. Together they were among the first to seriously study Leibniz's differential calculus and became co-founders of the calculus of variations. Their collaboration turned into rivalry as Johann's talent matured, and by 1697 the relationship had completely broken down, with both brothers attacking each other in print.
Bernoulli's paper published in Acta Eruditorum in May 1690 is significant in the history of calculus because it contains the earliest known use of the word "integral" with its modern mathematical meaning. In the paper, Bernoulli showed that the isochrone problem was equivalent to solving a first-order nonlinear differential equation, which he solved using separation of variables.
Bernoulli had requested a logarithmic spiral engraved on his tombstone, along with the Latin motto Eadem mutata resurgo ("Although changed, I rise again the same"). The stonemasons instead engraved an Archimedean spiral, a different curve altogether. Bernoulli had written that the logarithmic spiral symbolized fortitude in adversity and the body's restoration to its perfect self after death.