I suppose you are asking about real radios (wireless). They were quite bulky decorated wooden boxes. They had vacuum tubes instead of transistors. They were considered (somewhat) luxury items and not every family owned them.
They had long, medium and short waves ranges. Later ultra-short (AM and FM were added). First transistor radios appeared in 1960s. They were small and portable and gradually displaced the old big boxes. Here is a typical picture:
Here is an old Soviet advertisement (probably mid 1960s) showing
two stationary receivers and an early portable (transistor) one:
The big item on the left is a "radiola", a radio combined with
vinyl disk players. And the one in the middle is a "magnetola", a radio
combined with a tape player. They were more common than just radios.
The item on the right is one of the two very first Soviet portable transistor receivers.
Let me add that the word "radio" also had another meaning in Soviet Union.
All Soviet homes were equipped with cable "radios", which were just small loudspeakers connected to wiring. Most of them had no amplifier, only a loudspeaker. Wiring for them was provided in ALL residential buildings.
In most cases only one program was available, so the "radio" had only one button which controlled volume, and nothing else. The broadcast started early in the morning
with the anthem of Soviet Union, and ended at midnight with the same.
They broadcasted news, political speeches, musics, radio plays, all sorts of useful information etc. This thing was in every home and almost all people listened to it. The program was the same everywhere (broadcasted from Moscow),
with short inserts of local news.
Абонентская радиоточка - Russian Radios (Wikipedia)
On the other hand, many people who owned a real (wireless) radio listened foreign stations (BBC, Voice of America, Free Europe, and many others which broadcasted in Russian). As I understand this was one of the main reasons to buy
a true radio receiver. As a child (in 1960s) I even wondered why they make receivers with short waves. It seemed to me that the only use for short waves was to
listen foreign stations:-)
In the late 60s Soviets started to jam these broadcasts,
but they were not very successful.
Let me also add that "amateur radio" was a very common hobby: people made receivers and even transmitters themselves. (One had to register with the authorities every transmitter). All sorts of parts and literature were widely available in hobby stores in the 1960s.
EDIT. It is certainly not true in 1960s that "every house had a real radio receiver with an antenna". Cable "radios" in houses were connected by cables to a place called
"radio-node" (радиоузел in Russian), one or several in a city, depending on the size of the city. How were the radio-nodes connected with Moscow, I do not know. And in 1960s there were no channels and switches on these "radios", They were introduced later.
Here is a nice collection, complete with authentic sound, both wireless and wire.