China is a very complicated example. Up until 1980, it was a classic planned economy that had all the inefficiency problems of a planned economy, compounded by the failure of Maoism as an industrial policy. For instance, instead of building several large steel mills, Mao thought it would be better to issue tiny little blacksmith forges to all the peasants, which they could use to forge small quantities of steel when not working the fields. Kudos for trying to leverage China's biggest asset (its insanely large population) but making steel isn't like planting rice. It takes a certain amount of skill and quality of materials and the resulting output had massive quality control issues.
After Mao's death, Deng Xiaoping immediately took steps to move the country towards a more moderate economic system. The Shenzhen special economic zone was created as a laboratory of sorts to see how capitalism works. They took a tiny little fishing village just across the water from Hong Kong and said "Ok, in this zone we'll allow private investment and private ownership (but with the caveat that the government gets a partial stake in all companies)." It went gangbusters. Today, Shenzhen alone has a GDP of around $145 billion. They expanded the program to other cities (Guangzhou, Tianjin and Shanghai being the biggest ones) and in 1990 they opened the Pudong zone in Shanghai to foreign investment.
The areas of China outside the SEZ's is still more or less a command economy, and it's still majorly underdeveloped. The answer seems to be "add more SEZ's", but the rich/poor divide between the coastal cities where the bulk of the zones are located and the interior cities which are mostly lagging behind is a MAJOR (probably the major) issue in China today. It's created a huge internal migration problem, it's breeding social tension, and it's still a barrier to further growth. Eventually, I think most of the country will be in an SEZ of some form or another, and the country will essentially be state capitalist.
But here's the other thing going on. While the major industries are all state capitalist, where there's private investment but the government maintains a sizable interest stake in the company, there are an insane number of smaller businesses that the government doesn't have a stake in and which are difficult to regulate. In some ways, it's like the peasant-forged steel problem all over again. Lead-based paint on toys? Melamine in the milk? Poisoned cough syrup? Those problems aren't because the government of the PRC is corrupt, it's because it's damn near impossible for them to regulate when there might be 6,000 different toy making companies in Guangzhou alone. The free market competition between them is intense, which encourages cutting corners to lower costs and outdo the competition. Workers' rights in many of the SEZ's are almost nonexistent (a very bitter irony), and in a place like Guangzhou or Shanghai, you regularly see people who were horribly maimed in industrial accidents (in part because employers aren't providing safety gear or training, and there's no vigorous government regulation akin to OSHA) and employers regard laborers as expendable because there are a constant flow of impoverished liudong ren ("floating people", the illegal internal migrants from the non-capitalist interior) lined up to fill those jobs. So these crippled wrecks of people are left to beg in the gleaming streets and subways, while fellow Chinese in Gucci suits and driving Escalades step over them or maybe toss them a yuan or two. It's not dissimilar to say, New York at the turn of the 20th century when you had ridiculously wealthy guys like Vanderbilt and Carnegie whose wealth was built on the backs of immigrant laborers who lived in squalor and couldn't afford to even set foot in the gleaming towers that they built. Bottom line, capitalism is destroying just as many lives in modern China as Mao's poorly planned economy did in the 1960's.
As far as the government, it's Communist, but it's not (and never has been) like Russian Communism. Chinese Communism has always been distinctly Chinese. Which is to say that it's really not that different in many ways from the various Imperial bureaucracies that preceded it for the last 2000+ years. Authority is concentrated in a few individuals with a single prominent leader, but actual power is decentralized because it's damn near impossible to run a country that big from Beijing. Much of the actual power in China is at the local and provincial levels. Beijing doesn't get involved in provincial affairs unless there's a problem. There are LOTS of problems, but the local and provincial governnments do their best to hide this, because they essentially have their own pocket kingdoms and like it that way, and don't want the higher authorities to come in and investigate and possibly take their kingdom away from them. Again, this isn't a problem with Communism, it's a problem with being China. The corruption of local officials (especially the "hiding it from the Emperor/King/Prime Minister/Premier/etc" part) has been a trope in Chinese literature, art and film since before the First Emperor. It's also one of the biggest obstacles to effective regulation and reform. IMHO, Beijing is honestly trying to clean up corruption and to implement as many 'green' technologies and policies as they can. They don't need to worry about slowing down economic growth, because they WANT to slow down economic growth. The central Party economists have realized that double-digit growth rates in GDP every year is simply not sustainable, especially when it's still so geographically unbalanced. But local officials often look for every excuse to ignore orders because it would negatively impact the bribes and kickbacks they've become accustomed to.
I guess what I'm saying is that China is not a good example to judge either communism or capitalism by, because China is simultaneously both and neither.