What are the inventions of the 1920s and how do they relate to The Great Gatsby?

Probably the most important modern invention to The Great Gatsby is the automobile. The automobile was invented in the 1890s, and the first Model-T came out in 1908, bringing cars to the average person before the first World War, but it was in 1920 that car ownership topped one million. The Roaring Twenties are associated with the widespread use of the automobile and the fast-paced lifestyle it brought.
Automobiles are important to the plot of The Great Gatsby: the pretense of having a car to sell gives Tom the excuse to call the Wilsons' garage as well as to stop in, making it easier for him to set up meetings with Myrtle. Beyond that, Gatsby's long yellow car characterizes him as a thoroughly modern, self-made man. Showing he attracts a modern crowd, long lines of cars arrive at his mansion for his wild, jazz-age parties. It's notable, too, that while the patrician, upper-class Tom has a string of polo ponies, Gatsby doesn't own a horse, only car. Most significantly, it is Gatsby's huge car that runs over Myrtle, setting the tragic ending of the novel in motion.
The telephone is another invention that predates the 1920s, but widespread phone use actually doesn't begin to take off until the post-World War I period. In Gatsby people communicate by phone all the time, although many people in the US wouldn't have phones until the 1930s. To some extent, the phone shows the privilege of the people in the novel—or in Wilson's case, the need to have one for a business.
Built-in swimming pools also predate the Roaring Twenties but are often associated with that period and the economic excess of the boom years. Gatsby's pool is important to his party-goers, although he doesn't use it until the last day of summer, and then he is killed in it. It represents both his wealth and the fact that his goods were not purchased for his own physical pleasure but in pursuit of an elusive dream.