Did people receive free electricity in their homes in the Soviet Union? If so, how was it rationed out? Were there any quotas on how much you could use?
No, individuals didn't have and couldn't have electricity for free. It's clear that if there were just a limit on the maximum consumption of electricity (or water or anything else), most people would saturate it and the production wouldn't be sufficient.
The prices only followed the template of "total regulation" and "permanent monopoly", so they were constant for a very long time and universal everywhere in the whole country. Because of the absence of the market, no one actually knew whether the electricity sector was subsidized or profitable.
Also, one must realize that most of the consumption wasn't attributed to households but to organizations (public transportation, factories, schools, hospitals etc.) and all of them were basically state-owned and got whatever subsidies they needed so all of their accounting (including the consumption of electricity) was a political issue, not a financial issue.
See e.g. the book Electricity Market Reforms: Economics and Policy Challenges
Electricity was not free for the population, as Matt said, the price was 4 kopecks for 1 kW·hr since 1961 to the end of 1980s. Luboš Motl said the price was everywhere the same, but I am not sure. There could be variations regarding the "belt", a term used for "price belts", the zones which had different prices in the USSR (such as Far East, North etc). Also, those who had no natural gas pipeline at home and used electric ovens had cheaper electricity, at 2 kop per kW·hr.
Some categories of people (such as WWII veterans etc) could have discounts, but I do not know the details.
Each apartment was equipped with an electric counter. The shared flats had a counter for each room and a counter for common area.
Note also that there were no water counters, nor gas counters so that one could use as much water and/or gas as one wanted, the price was fixed per month.
There were no caps that I am aware of.
As to industrial consumers, they paid completely different non-cash money for electricity and other utilities and supplies, and the prices for them were totally different.