The soviets themselves were Councils, think of them as political units. You can find a brief essay by Soviet Historian Lewis Siegelbaum on them here: http://soviethistory.macalester.edu/index.php?page=subject&SubjectID=1917formsoviets&Year=1917

(It's an excellent site for Soviet History in general, as a side note)

First it is important to remember that there were two Revolutions in 1917. In February, the Tsar was overthrown and replaced by the Provisional Government. It was in February that the Soviets also began to organize as a mode of democratic (at least in rhetoric, more on this in a minute) mode of rule. So, in the aftermath of February there were competing forms of government.

By the time October came around, after the July Days and other unrest and dissatisfaction with the rule of the Provisional Government, the Soviets represented an alternative for popular rule. So when Lenin said "All Power to the Soviets!" in July 1917, it was the a challenge to the Provisional Government by calling for a formal transfer of power to the (socialist dominated) Soviets. Although the soviets were city based, it is worth realizing that delegates and members from organizations like trade unions were involved - things that we don't normally associate with political positions per se. They did have basically democratic ideas - elections, representatives, delegates, and so forth.

After the Revolution, a formal structure was established that went from local on up the All-Russian Congress. It was not nearly so easy in reality, particularly in light of the fact that although this structure was established in 1918, the Russian Civil War continued on for several more years - and contained the political chaos you'd expect from such an event.

More on that here: http://soviethistory.macalester.edu/index.php?page=subject&SubjectID=1917soviets&Year=1917&navi=byYear

Lenin wrote:

Democracy is the rule of the majority. As long as the will of the majority was not clear, as long as it was possible to make it out to be unclear, at least with a grain of plausibility, the people were offered a counter-revolutionary bourgeois government disguised as "democratic." But this delay could not last long. During the several months that have passed since February 27 the will of the majority of the workers and peasants, of the overwhelming majority of the country’s population, has become clear in more than a general sense. Their will has found expression in mass organisations—the Soviet’s of Workers’, Soldiers’ and Peasants’ Deputies.

(https://www.marxists.org/archive/lenin/works/1917/jul/18.htm)

The Soviet Union as a name is a little less descriptive than the full name - in russian Сою́з Сове́тских Социалисти́ческих Респу́блик (Soyuz Sovetskikh Sotsialisticheskikh Respublik) the Union of Soviet Socialist Republics. The idea being to unify the soviets which had been established across Russia as a single socialist/communist government. Indeed, the second and third All-Russian Congress of Soviets happened within days and then months of October respectively.

From the Third All-Russian Congress of Soviets: http://soviethistory.macalester.edu/images/Large/1917/aih528.jpg