Higher education was free, as was much else. The arrangement was part of the standard Communist contract with society: “stay obedient, stay poor, get things free”.
Meritocratic selection
As the number of admissions was limited, you needed grades to pass. In my time (the 1970s and 1980s), it was two steps: first, you needed an average grade from the last year in school, and then you get an average grade in the admission exams (typically 4–6 subjects testing what you learned in school). The ones with the highest average get past the post.
The two-step admission existed because universities didn´t trust school grades. Standards of teaching varied greatly, and most of time you could simply buy better school grades. You could as well bribe your way through the admission too, but it was more expensive and complicated (go find what palm to grease if you are a 17-year old farmer boy first time in a big city).
Spots of corruption
Were you an offspring from some big shot’s family, all it took was a series of telephone calls through the friends-and-family network down to the admission officers. That kind of non-monetary admission was much preferred. Bribery was officially frowned upon, and everyone felt great about helping their pals. The true Age of Innocence!
Ideological footprint
The faculties were basically similar to the largest Western universities. The curriculum was very conservative (often lagged behind the Western one by 10–20 years), and typically included at least one Communist indoctrination course, as well as a mandatory introduction to every subject, e.g. “The role of the Communist Party guidelines from the XXV Congress of the CPSU for further development of particle physics in the USSR”.
Master’s grade was the standard, and required at least 5 years. Grading system was 1–5, with 5 the highest. Most of students managed 4s and 5s, the sloppy ones got the passing 3. Never heard of anyone of my friends who´d ever managed 2.
Learning by rote
Learning method was the glorious cramming of the curriculum, inherited from 19th century Prussian universities. Memorize as much as you can from the professor´s lectures and the books, and you get your grade. Some islands of creative teaching and university research produced world-class graduates, mostly in sciences: the Soviet military needed bright minds and smart weapons. But they were few.
Throughout the Soviet era, the teaching standards were more and more following the international level, and intellectual curiosity was gaining ground. The problem was that the older generation of professors and researchers — those who got their education from the hands of good old masters from pre-Communist times before the indoctrination and rejection of “bourgeois science” — these were retiring. The new academia were a complete product of Soviet era that valued conformity and following orders above curiosity and critical thinking.