This is a HUGE topic. I'll try my best to give a comprehensive summary.

I'll start with the end of WWI. The Treaty of Versailles was signed on June 28, 1919. This treaty ended the state of war between Germany and the Allied Powers (US, France, Britain). Each of the Allied Powers had their own aims for the Treaty of Versailles. I'll briefly outline them:

• French Aims - Wanted to weaken Germany as much as possible since they shared a boarder. The French Prime Minister wanted to either extend the French boarder to the Rhine or create a buffer state in the Rhineland. This didn't happen. Instead the treaty forced the demilitarization of the German Rhineland. Also, they supported heavy payment reparations. The Germans had to pay the Allies for their losses (both human and land). There was obviously more that the treaty demanded, but this is the most important (in my opinion).
• American Aims - Wilson was concerned with rebuilding the European economy, encouraging self-determination, promoting free trade, creating appropriate mandates for former colonies, and above all, creating a powerful League of Nations that would ensure the peace. He opposed harsh treatment of Germany but was outmanoeuvered by Britain and France.
• British Aims- Britain suffered little land devastation so therefore they they supported reparations to a lesser extent than the French. The British wanted Germany to be a strong trading partner down the road and they were worried that heavy reparations would harm that.

So, the results of the Treaty of Versailles was a demilitarized Germany and a Germany that had to pay 132 billion Marks ($442 billion in today's money) in reparations. This caused a HUGE stress on the Germany economy and with the world's economy drained by a 4 year war, the Great Depression started.

The economic troubles of the 1930s were worldwide in scope and effect. Economic instability led to political instability in many parts of the world. Political chaos, in turn, gave rise to dictatorial regimes such as Adolf Hitler's in Germany and the military's in Japan. (Totalitarian regimes in the Soviet Union and Italy predated the depression.) Also, the terms of the Treaty of Versailles left a sour taste in Germany's mouth. In come the Nazis. They campaigned on the idea of bringing Germany back to her glorious self. They wanted to undo the humiliation of the Treaty of Versailles. They started militarizing and occupying areas that were forbidden by the Treaty of Versailles (the Rhineland). Germany and Russia sign a non-aggression pact, Germany invades Poland on false pretenses, Britain tells Germany to cease military operations, they don't, and WWII starts. (There's obviously more to this, but your question asks how WWII is related to WWI and the Great Depression).