The Great Depression was a period in the early 20th century on Earth in which the entire world experienced an economic downturn. It was focused mainly in North America - specifically the United States and Europe, but it affected the entire world. The downturn also brought about severe social upheaveal, with large numbers of people being unemployed, and a large number of homeless people as well.
The Depression began in the late 1920s, most have identified the stock market crash of 1929 as the beginning of the Depression, and it lasted throughout the 1930s. Franklin D. Roosevelt was elected in 1932 on a promise to introduce reforms to help get the U.S. through the Depression. A number of public works projects were started to help get the population to work. One of these projects that still existed in the 23rd century was the Mount Rushmore National Memorial. For over a decade, Roosevelt led the United States through the Depression and the Second World War that followed it until his death in 1945.
In the 23rd century, Doctor McCoy suffered an accidental overdose of cordrazine. He escaped from the USS Enterprise to the planet Gateway. After following him down, a landing party led by Captain James T. Kirk and Mr. Spock encountered the Guardian of Forever. McCoy jumped through the Guardian, winding up in New York City in January 1930. While in the 20th century, McCoy managed to change human history, resulting in the Federation never coming into existence. Kirk and Spock followed McCoy back. They rescued McCoy and restored the timeline. (TOS episode: "The City on the Edge of Forever"; miniseries: Crucible)