It seems like European nations are all establishing colonies, but I don't understand why China would be any different. It seems like it has the power and resources to do so numerous times throughout history, but they never do. Why is this?
They did. Depending on the preferred definition of "colonies", Chinese states in fact established innumerable colonies throughout history. Certainly the most common form was overland colonies created in conquered "barbarian" territories. This processes lasts up till today; Beijing's sinicisation and settlement policies in Tibet and Xinjiang are viewed with some justification as colonialisation.
While not overseas like how we usually envision "colonies", this isn't unique to China. In Europe, the colonial expansion observed in Tsarist Russia similarly spread overland. In the earliest times, there's also a certain similarity to the Roman and Greek colonies of classical antiquity. More stereotypical colonies were also founded later, though in most cases they were eclipsed and swept away by the more successfully European colonial empires.
See below for some examples on the differing forms of Chinese colonialism.
The Zhou tribe from modern Western China conquered the Shang Dynasty circa 1046 BC. One of the first recorded acts of the new dynasty was to dispatch members of the royal family and other loyal supporters throughout the Central Plains. This took the form of armed settlers establishing garrisoned cities, from which natives were ruled by loyal Zhou vassals. These were colonies much in the same manner as the Roman Republic's coloniae.
(Map of the Early Zhou Dynasty. Circled dots represent "states" established by the Zhou royal court. These were similar in nature to the cities states (or colonies thereof) of Europe during classical antiquity.)
Originally, the Zhou dynasty was largely a network of city states exerting loose control over territory in between. Barbarian tribes dominated much of this space, and at times became serious existential threats to the Chinese states. Most of these, especially near the centre of Zhou civilisation, were gradually subdued, colonised, and assimilated. By the Warring States period, the external borders of Chinese civilisation had made dramatic advances.
(Not explicitly shown is the expansion of the states into previously encircled barbarian lands)
Notice the northeastern expansion of Yan around modern Liaodong and into Korea; the southern expansion of Chu towards Vietnam; the northern expansion of Zhao towards the steppes; and the dramatic expansion of Qin in pretty much all directions. This was colonisation, in the same way that Russia colonised Eurasia.
Additionally, Chinese settlers established lasting colonies in the Pacific, such as Hainan or Taiwan. These were very much comparable to the overseas settler colonies of Western European powers, such as the Spanish and Portuguese in South America, or Britain in Oceania and North America.
Lastly, although Chinese commercialism were traditionally restricted by prejudice, it eventually began to flourish. Overseas trade became active (albeit with ups and downs) during the Song and Ming dynasties. By the early modern period, some Chinese settlements began to take shape in South East Asia. Examples include Sulu, and parts of modern Malaysia or Indonesia.
Apparently, some Chinese also settled o northern Borneo and perhaps also in the Sulu area ... there was also a rather large Chinese group in the Palembang region.
Guillot, Claude, Denys Lombard, and Roderich Ptak, eds. From the Mediterranean to the China sea: miscellaneous notes. Otto Harrassowitz Verlag, 1998.
These were more similar in nature to the trade posts established by European maritime powers during the first wave.
Notice how big China is? There's a reason for that; it's only a semantic difference between calling conquered territory a "colony" and simply part of your country.
EDIT: Someone pointed out in comments that the term "colonize" means something different from "expanding borders". So I should clarify what I mean: yes the terms are different, but it's just a semantic difference. If China were limited in how much it could expand (eg. it's an island, like Britain) then it would have "expanded" through colonizing, but they didn't have to. The only real difference between colonizing and expanding is whether or not the conquered territory shares a border.
China did have colonies. All of the islands in Asia reachable by junk have been colonized by the Chinese at one time or another: Malaysia, the Phillipines, Taiwan, etc. The far ranging colonies of the European powers made in the 1500-1800 period cannot be compared because China did not have types of sea-faring vessels necessary.
Another factor is that China lacked a cohesive zealotry that is often the impetus behind colonization. For example, the Arabs colonized many places, including southeast Asia during the Islamic expansion, but this may be attributed to their religious zealotry. In China, a place having many different nations, peoples and languages this kind of force did not exist. Also, one of the most common religions in China, Buddhism is a passivistic religion that does not encourage conquest and attack.
In the early 15th century, China had huge junks that dwarfed the ships of their European counterparts. China's Treasure Fleet sailed throughout the eastern Pacific and northern Indian oceans. By the latter part of the 15th century, China had turned inward. Building or working on a junk with more than two masts became a capital crime.
Who knows what the world would look like had China not taken that turn away from science and exploration in the early 15th century to court intrigue by the end of the 15th century. It would make for a good alternate history novel.
If you count areas occupied by China that has less than a majority of Han Chinese people, either today or at least when first claimed, then China has numerous colonies along its western borders; Sinkiang, Tibet, parts of Mongolia, etc.
The reason why China is not considered "colonialist" in the usual sense of word is because historically, it has NOT had Chinese settlements in lands that were "noncontiguous" to China.
For instance, during the Ming dynasty, Admiral Zheng He (and numerous admirals before him), visited parts of modern Indonesia, India, the Middle East and Africa, (some say, even western North America), but did not establish Chinese settlements in these places or try to conquer them. This is in contrast to European countries that later did conquer these areas.
After reading the interesting explanations, it strikes me that there were at least two reasons why the western European nations, at least those with significant maritime capabilities, sought to colonize: (1) they had no significant military advantages over the other nearby nations. Colonial expansion into the neighboring countries would have been a costly proposition. (2) If everything comes down to money, then perhaps the desire to colonize was based more on improving their financial strength than invasion and occupation; it was all about the benjamins.
Tom au comment on South China Sea is very important to understand the reasoning. Why would China colonize South China Sea when there are so many other islands? The answer is simple, because China is interested in trade only.
I recently visited Philippines and found that China had trade relationship with Philippines before the arrival of Spanish/Portuguese. They even collected some form of revenue from the Estates or loosely taxes but never gave the islands their name or religion or language. I think with China being so large and diverse they had respect for another region/ island for their language /culture etc and did not want to convert them to be like themselves. Yes they did settle wherever they went to trade.
Same logic applies with India also being such a large and ancient civilization they spread slowly through trade routes only not through religion/ language and culture domination. Whatever India spread through religion in Indonesia eg Bali or Cambodia Ankor Wat was primarily through trade routes and was spread by the religious people who were part of the tradesman rather than the ruler.