The Great Depression is and will always be a monetary phenomena. Monetary policy caused it, ended it, but didn't decrease unemployment down to frictional levels. Herbert Hoover, who was president during the initial phase of the depression, fervently stuck with the gold standard and defended the policy of sound money which only exacerbated the economic downturn. Many countries started to recover in the mid 1930's when they dropped the gold standard - allowing a greater expansion of monetary policy. Economists and historians find that the sooner a country dropped the gold standard, the sooner they recovered.
In his article and book "The Midas Parodox", Scott Sumner describes the impact of the international gold market on monetary policy as:
P * Y = P_g * g_s / (r * m_d)
Where P is the price level, Y is output, P_g is the nominal price of gold, g_s is the monetary gold stock, r is the gold-reserve ratio, and m_d is the quantity of money demanded by market actors.
We can view P*Y as a collection of income and expenditures, or, simply, NGDP. P_g is exgenously chosen by government. r is mostly determines by the central bank and P_g is endogenous. The monetary gold stock is the quantity of gold held by the central bank in vaults. The non-monetary gold stock is the quantity of gold held in the private sector. m_d can be influenced by many things, but if market actors want to hold more money then they will bid up the price of money, which is defined as 1/(P*Y).
During the Depression, we saw Shocks to all those variables that lead to an increase in P and a decrease in Y. There was also a massive amount of gold hoarding. The US and, especially, the French central banks were also increasing their gold reserve ratios by simply our basing more gold and also by increasing reserve requirements. We can infer that m_d was rising as well, but it's harder to do this in quantitative terms due to data limitations. Generally, m_d tends to be inversely proportional to money velocity, but that relation doesn't ways hold. Finally, the various labor market interventions during the second new deal led to an increase in P. In order to cancel out the effect of all changes, Y must decline - which means a recession is happening.
What finally got us out of the Great Depression, in terms of GDP which is appropriate for a manufacturing/industrializing economy, was an increase in P_g when FDR re-pegged the gold to US dollar exchange rate. I said in terms of GDP on purpose, because although GDP was returning to normal levels, the unemployment rate was still staggeringly high. This is where, what is called a "natural test", of quasi-keynesian economics comes into play.
Keynesian economics is the argument that the amount of total spending in the economy can affect the amount of economic activity in the country. e.g. if the government decides to buy more stuff that month, more will be produced.(Fiscal Multiplier)
What eradicated all traces of the Great Depression ( the unemployment side), was an increase in military spending during World War II, that acted as a stimulus for the economy.