China is a term used to refer to two countries; the People's Republic of China comprising a large part of eastern Asia, and the Republic of China, more commonly known as Taiwan, almost exclusively comprising the island of the same name off Asia's Pacific coast.
 The one on the continent
Mainland China, formally the People's Republic of China, is the world's second largest country. It has the largest population of any country in the world. China has a well documented and long history, has produced many philosophical traditions, and has recently grown to become a major power 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 harsh government-sponsored programs that devastated China, such as the "Great Leap Forward" and the "Cultural Revolution." A conservatively-estimated 20 million people died as a result of these two programs.
The current government is nominally Communist, but is in reality functionally capitalist; for instance, China is currently privatizing its agricultural sector. Basically, China is maintaining its authoritarian government while having a fairly laissez-faire economic system (great job, Milton).
A province (shěng or 省), formally provincial level administrative division, is the highest-level Chinese administrative division. There are 34 such divisions, classified as 22 provinces, 4 municipalities (huge metropolitan cities), 5 autonomous regions (as big as provinces but has ruling body and laws tailored for local ethnicities other than Han), 2 Special Administrative Regions (Hongkong and Macau, cities that remained under lease until the late 90s), and Taiwan the "little".
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" the Republic of China (Taiwan) from what used to be called a "fascist Western puppet state" and finish the civil war.
 The one on the island
In the aftermath of the Communist revolution's success on the mainland in 1949, the remaining Nationalist counterrevolutionaries from the defeated Kuomintang (KMT) forces escaped to the island of 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 outer Mongolia and Tuva. The ROC currently only 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. People on the mainland are split between supporting reunification and declaring independence. Those who support reunification (known as Pan-Blue, or supporters of the Kuomintang party), however, disagree as to which government should rule over the unified China. Supporters of Pan-Green (the more liberal party) support abandoning the One-China policy and renouncing all claims on the mainland (ie. proclaiming an independent Taiwan). Regardless of their views, however, most people in Taiwan can agree that they view themselves as Taiwanese, not Mainland Chinese (which is also an insult for a plethora of things in Taiwan).