Hydra has a hierarchical command structure with an individual leader in charge of all operations. The organization is broken into cells working on different projects, with limited contact between cells to limit overall damage if one cell is discovered and destroyed. Redundancy is built into the organization at all levels; their motto is "If one head is cut off, two more will rise to take its place."
Leading up to World War II, the organization was subordinate to the Nazi military command in Berlin, but Schmidt eventually took the organization rogue.
After Schmidt acquired an alien artifact called the Tesseract, he took the organization rogue in an attempt to use its power to gain control of the world for himself. This plan eventually failed, and Hydra was considered dead.
One of Hydra's top engineers, Arnim Zola, was recruited by the incipient SHIELD organization as a technology advisor. With his recommendations, various other Hydra sleeper agents infiltrated SHIELD and slowly worked their way into the highest levels of that organization.
In 2014, Hydra agents staged a coup to take over SHIELD and seize world power using SHIELD's new fleet of helicarriers to eliminate all effective opposition in a massive first strike. The attempt failed, but SHIELD itself was reduced to a tiny fraction of its former influence, and hidden Hydra cells remain active.