Even though the Republic of China (ROC, also known as Taiwan) is not a member of the United Nations, it has been economically successful. What is the secret behind the economic success of it then?
One word - manufacturing. Before China was the country of cheap labor it was Taiwan, just as Japan was before Taiwan. Taiwan built up a lot of manufacturing know-how, and still does so in computer chips and some other respects, it also has many companies that manufacture on the mainland. The capability of Taiwan being able to do this also coincides with its economic system, which as differentiated from the Maoist-Communist system on the mainland, allowed entrepreneurship to blossom as Japanese and US investment came into Taiwan.
For a recent economic overview on Taiwan's GDP Growth:
The Gross Domestic Product (GDP) in Taiwan expanded 1.06 percent in the first quarter of 2012 over the previous quarter. Historically, from 1981 until 2012, Taiwan GDP Growth Rate averaged 1.4000 Percent reaching an all time high of 5.6400 Percent in December of 1990 and a record low of -5.0700 Percent in December of 2008. The Gross Domestic Product (GDP) growth rate provides an aggregated measure of changes in value of the goods and services produced by an economy. Taiwan is an industrialized, developed country, just off the coast of China. Taiwan's economy, one of the "Four Asian Tigers", is export-oriented and specialized in production of electronics and machinery. In fact, Taiwan is one of the world's largest suppliers of computer chips, LCD panels, DRAM computer memory, networking equipment, and consumer electronics. Textile production, although already in decline, is another major industrial export sector. This page includes a chart with historical data for Taiwan GDP Growth Rate.
And for a historical review of Industrial Development in Taiwan you can review Digital Taiwan's overview which puts it better than I can to cover their many areas of Production.
Edited: So I guess if you ask for the Why then you can look up the Taiwan Miracle where although land reform moved labor to Urban Centers, providing for a cheap factory worker group you also have increased US Aid which
created a massive industrial infrastructure, communications, and developed the educational system
Without that industrial infrastructure you could not have the manufacturing take off that the country had, in my opinion. Again the economic system plays a very important role, Capitalist markets on Taiwan allowed it to grow and become a manufacturing/exporter with a cheap, educated labor pool. Since many of these reforms, or a cheap manufacturing workforce, never took place on the Mainland until Deng Xiaoping began reforms that set the stage for the China we have today. As noted on Suite 101's Chinese History page Deng began reforms in 1978 called the
gaige kaifang, or “reform and opening up.” These were intended to jumpstart China’s economy and caused rapid economic growth in China.
Once the commune system was abolished and the markets began to open then China became a place to invest and manufacture in. There is a good overview of Deng's reforms that note them in detail. But without the market forces that began to operate on the mainland in agriculture and industry, just as they did in Taiwan previously, then China never would have taken over.
Although considering the state of Taiwan for the past 10 years I am not sure you can call them an economic success anymore, considering their marginalization within the UN world body they do pretty well. As to the marginalization in numerous world bodies that is another question in itself that deserves its own answer. And yes I know they still do well as a county, my wife comes from Taiwan so I keep an eye as to what is going on there.
One East Asian economic miracle occurred in Japan. A few years later 4 more occurred in S. Korea, Taiwan, Singapore and Hong Kong.
All of these countries and Hong Kong had a few things in common.
All had spent significant time being governed by Either Japan or England. None were democratic and so were able to avoid the temptations of socialism. All managed to avoid a communist revolution for various reasons. All were cultures that prized education.
As for Taiwan in particular, they were rebuilding their wealth, not just creating it from scratch. The Japanese has built infrastructure and schools while ruling Taiwan. Just before WWII Taiwan was the wealthiest region in Asia outside of the Japanese home islands. After the war the Chinese looted Taiwan, introduced corruption and caused unrest, but the human capital (an educated experienced workforce) and much of the infrastructure survived.