The Soviet structure changed multiple times in history. I’m going to talk about the pre-1989 system. There’s a lot of really weird “communist” administrative names that get used, so it gets pretty confusing.
The Soviet system is based around the idea of “soviets”, which roughly means workers’ council. Furthermore, the administrative system is split between the actual government and the Communist Party.
Rurally, people would vote for their village soviet (city council). Each village soviet would send a delegate to the township soviet (county council). The township soviet makes laws for that particular area.
In cities, it was slightly different. People from different productive groups (unions) would send delegates to the city soviet (city council).
It’s insanely complicated at the provincial/district level, but the idea is the same. Local councils send delegates to higher-up councils. So forth.
At the very top, you had the Supreme Soviet (House of Representatives). These guys were supposedly the highest legislative body, but were really just rubberstamping whatever the Communist Party wanted. They also selected the Council of Ministers, which were the guys running the day-to-day operations (education, infrastructure, etc.). The head of the Council of Ministers was the Premier of the Soviet Union.
In reality, the country was run by the policymakers internal to the Communist Party (CPSU). These policies were supposedly created by the Congress of the CPSU, which was composed of delegates from around the USSR.
However, the Congress only met every few years, so most of the actual decisions were made by the Central Committee, which was separated into the Politburo and the Secretariat. The Central Committee also included other members, but was often only rubberstamping what the Politburo wanted.
The Politburo were the head honchos. They made the big policy decisions. Most people think of the Politburo when they think of the guys who worked with Stalin, Khrushchev, or Brezhnev. These are the guys who run the show, but you can see a lot of historical conflict between the Politburo and other organs of the government.
The Secretariat were the administrators responsible for the day-to-day running of the Communist Party. The leader of the Secretariat was the General Secretary and was the head of the whole CPSU. When we talk about “leaders of the Soviet Union”, we mean the General Secretary. Khrushchev, Stalin, Lenin, Gorbachev were all General Secretaries.
All in all, the Soviet government is really, really confusing. Especially when you realize that most of the “councils” and “organs” were rubberstamping orders from top-down.