...huh?
On paper, it works a lot like any other modern government. Local polities elect (no universal franchise: early on, the electorate was about 10% of the populace, by the end it was about 35%) members to the Central Committee/Congress of People's Deputies, which then selects the members of the Council of Ministers (an upper house of sorts), which determines the members of the Presidium (essentially a Cabinet), whose head is the President/Supreme Soviet. There is an independent judiciary with a supreme court. That's near the end; it changed a lot throughout its history. One version of the Soviet Constitution I;ve seen was 55 pages long - they change it often. It's theoretically federal in structure, with the Republics corresponding to US States.
In practice, the Party Chairman runs the civilian government, and there's always an uneasy three-way dance between the civilian government, the Red Army and the KGB. The Republics are meaningless as decisions are made at the center which they cannot fight. The Central Committee is too large to legislate constructively - 2,250 members at the end - so it meets briefly, appoints a single person or small group of people to deal with a problem, and then fails to oversee those appointees meaningfully. The Council of Ministers has a little over 100 members at the end, so it could have been a legislature, except it didn't try.