I know that the policy was more fiercely enforced during the early years of it's implementation, and probably got more relaxed as time went on until the policy was eventually abolished.
I am particularly interested because my wife was born in the 80s in Fuzhou, China and she has a brother while the 1 child policy was supposedly in effect. When I asked her about it, she told me she thinks her parents paid a fine and that was it. Which I found a little bit curious.
Your wife was born in the exact era when it just happened to be implemented. There several ways that her parents could have had more than one child, but seeing as she is both female and her parents had another child while paying the fine, I can quite safely draw the assumption that she was probably from a more affluent family at the time. Read more in detail below.
The "one-child policy" was initially introduced to address concerns regarding the alarming growth of the mainland Chinese population from 1949 till 1976, with the population spiking by 400 million, from 540 million to 940 million. The Communist Party of China (CPC) was concerned that such high growth would drain food/water resources and put the country under heavy economic, environmental, and social problems.
From the onset, however, the one-child policy's ideology was somewhat flawed. China's population boom can generally be attributed to three things:
A marked pattern of repetition whereby poorer, less-developed societies (as much of China was at the time) tend to have as many children as possible, as opposed to clever family planning
The population naturally "recovering" after coming out of a lengthy period of war and instability, ie. the American baby boomer generation.
A distinctly "Asian" family characteristic that encourages large and tightly-knit families (the "Confucian Family"). Additionally, the desire of Chinese families to have sons, and the value of sons over daughters, to the point where families would keep trying for more children until they had at least two sons.
These three factors, when combined, basically led to China's massive population boom. Such a population boom occurred in many places in the world (including the USA), but it was perhaps only in China that it was met with some alarm due to the country coming out of a prolonged period of war (far, far longer than any country at the time; China had been at war with others and itself since 1916), undertaking their own branch of 'Chinese communism' at the time, and wanting to really preserve their resources.
The one-child policy thus came into effect in 1979, but China is a vast country. Many of China's provinces are the size of entire foreign countries. The policy was thus most strictly enforced only in densely-populated, built-up city areas, where authorities could keep a closer eye on things simply by virtue of being able to monitor hospitals and other medical facilities where childbirths occurred. Women were forced to have intrauterine devices (IUDs) surgically installed after they had their first child, and they were installed in such a way that the IUDs could only be removed with a second surgery by medical professionals.
Ironically, these big, modernized Chinese cities were probably where the local population was more aware of family planning policies and were not the crux of the issue the Chinese authorities were hoping to tackle anyway. The problem lay in the countryside.
In the greater countryside, however, the one-child policy was much harder to implement. The peasant class often flouted the rules, and had as many children as they wanted. Officials rarely, if ever, checked up on some of the more isolated country settlements. Even if they did check, these peasant families would simply send their children off to hide somewhere until the officials departed.
It was also something of an open secret that wealthy, more affluent families living in the big cities could ignore the one-child policy by virtue of their wealth. If they wanted another child, they simply paid for the fine.
By the time the children born in 1979 and onward reached adulthood, the authorities were realizing a rather significant problem. The population of China was now badly skewed towards a male over female ratio (it was 111:100 males to females in 1990), as the Chinese families reasoned that if they could only have one child - may as well have a son, rather than daughter who will be married out to another family. Chinese males simply could not find marriage partners due to the larger male population, and many had to search overseas.
Furthermore, the desire for a son led to those families who had daughters to quickly give them up for adoption, thus "freeing up" their slot, so as to speak. The number of girls in state-sponsored orphanages thus went up drastically. Those who went to the hospitals prior to discover the gender of the child would often abort immediately if it was a female fetus. This act of selective abortion was primarily responsible for culling the female population of China. Collectively, the combined acts of selective abortion, and giving up female children for adoption, led to what is known as the "missing girls generation" of China.
Additionally, authorities failed to take into account several problems with reducing the population as much as they did, such as an ageing population in China that now does not have enough younger citizens to support the workforce in their absence and hold up the country's economy. This spells trouble not just for China, but for the world at large, due to the global manufacturing base being heavily concentrated in China.
By 2015, to address the mountain of growing problems, the government introduced the idea of a two-child policy, probably in the hopes of both allowing population growth and balancing out the skewed male to female ratio. In 2016, this policy came into effect, officially repealing the one-child policy and bringing it to an end.
TL;Dr version, the one-child policy was only really heavily enforced in the larger, modernized cities, where the authorities could keep a closer eye on things. In the countryside, it was next to impossible to implement, and the census records there are still hazy due to the many years of the one-child policy and the time spent by the people there hiding their 'extra' children. Wealthier families were also practically unaffected, as they could simply pay the fine and have however many children they wanted.
References:
Bongaarts, John, and Susan Greenhalgh. "An alternative to the one-child policy in China." Population and Development Review (1985): 585-617.
Fong, Vanessa L. Only hope: Coming of age under China's one-child policy. Stanford University Press, 2006.
Zhu, Wei Xing, Li Lu, and Therese Hesketh. "China’s excess males, sex selective abortion, and one child policy: analysis of data from 2005 national intercensus survey." Bmj 338 (2009): b1211.
Ebenstein, Avraham. "The “missing girls” of China and the unintended consequences of the one child policy." Journal of Human Resources 45.1 (2010): 87-115.
Zhang, Yuanting, and Franklin W. Goza. "Who will care for the elderly in China?: A review of the problems caused by China's one-child policy and their potential solutions." Journal of Aging Studies 20.2 (2006): 151-164.