Which parts of the Bretton Woods system were the most effective and why?
Though all points might be debatable from various perspectives, two parts of Bretton Woods, devised principally by economist John Maynard Keynes, that were significantly successful were the stabilization of world currencies and the creation of international banking institutions.
While there is plenty of reason to criticize the World Bank today (originally International Bank for Reconstruction and Development), the World Bank and the International Monetary Fund (IMF) did much to bring reconstruction to war torn and resource depleted countries which had been engaged in World War II. Global economic stability and increased global standards of living (including in underdeveloped nations, which are now developing nations) were tied to the operation of the WB and IMF.
Following the war, international currency prices were varied, effected by depleted coffers, strained by trying to rebuild bombed out cities and communities (some areas in Eastern Europe still today show the scars of war damage). Stability in world economies and world markets and expansion of world trade was critical to righting global financial markets after the crushing and destabilizing effects of World War II.