China is two countries, one comprising most of eastern Asia, the other an island off Asia's Pacific coast.
 The one on the continent
Mainland China, formally the People's Republic of China, is a big frickin' country with lots of people. It has a well documented and long history, has produced many philosophical traditions, and has recently been one of the big dogs in international politics, overcoming the results of Guns, Germs, and Steel, by which the European nations overpowered it militarily in the late 1800s. An extremist Communist named Mao Zedong ran the country for quite some time, and sponsored a number of government fiascos that devastated China, such as the "Great Leap Forward" (into their graves) and the "Cultural Revolution" (which was more like a Cultural Execution).
The current government is nominally Communist, but Karl Marx would turn over in his grave if he saw the way they run it. So would Joseph Stalin. This is not because China is performing too few executions, but because China is privatizing its agricultural sector. Basically, China is maintaining its authoritarian government while having a fairly laissez-faire economic system (great job, Milton).
Capitalism in China is particularly obvious to visitors: street merchants and vendor malls are regular sights, counterfeit goods are readily available, clothing stores sell alcoholic beverages, McDonald's and KFCs are large multilevel restaurants in major metropolitan areas, and haggling (to a certain degree) is expected. Green shops, or government-licensed stores, offer a "ten times your money back" guarantee on Chinese cultural merchandise discovered to be inauthentic, such as pearls, silk, jade, and tea.
There is lingering hostility between China and Japan over World War II and war crimes. Also, China is interested in eventually liberating Taiwan (see below) from fascists. Since the fascists gave up power almost 20 years ago, one can only conclude that fascist is a code word for "anti-communist" with them.
 The one on the island
In the aftermath of the Communist revolution's success on the mainland in 1949, a small group of loyal counterrevolutionaries from the defeated Kuomintang (KMT) forces escaped to the island of Formosa (Taiwan), where they have since maintained almost unbroken rule of the Republic of China (the one exception being 2000-2008, when Taiwan was led by the DPP's Chen Shuibian). Initially an oligarchy under Chiang Kai-shek, the ROC became a democracy after his death.
In accordance with the "One China" principle, the ROC continues to nominally claim authority over all of China, as well as Manchuria, Tibet, and Mongolia, since the Qing Dynasty was overthrown in 1912. Today, the ROC administers Taiwan and smaller outlying islands. Internationally, "Taiwan" is the common shorthand for the ROC, but the country is officially designated Chinese Taipei on the rare occasions when it can participate in international events (such as sporting competitions) alongside the PRC.
The international relations of the Chinas can get a bit sticky. Both the PRC and the ROC claim authority over all of China under the One-China policy. What this means is that other nations can only diplomatically recognize one or the other, and the vast majority of nations recognize the PRC; however, many Western governments have strong ties with Taiwan, including unofficial embassies. The PRC considers the territories controlled by the ROC to be a rogue province and gets very pissed off when foreign diplomats assert, or even imply, Taiwanese independence. Most people on the mainland support reunification, as do some on the island, but they disagree as to which government should rule over the unified China. Some on Taiwan support abandoning the One-China policy and renouncing all claims on the mainland (ie. proclaiming an independent Taiwan).
 The one in some American cities
Many non-Chinese cities have neighbourhoods known as "Chinatown" where people from the Chinese diaspora live and do business, often including selling specialty/imported foods and other goods from back home and running restaurants. In the English-speaking world, the best-known Chinatowns are probably those in New York City and London.