Neo-Confucianism was essentially a response by the Confucians to the dominance of the Daoists and Buddhists. Neo-Confucians such as Zhu Xi recognized that the Confucian system of the time did not include a thoroughgoing metaphysical system and so devised one. There were of course many competing views within the Neo-Confucian community, but overall, a system emerged that resembled both Buddhist and Daoist thought of the time and some of the ideas expressed in the Book of Changes (I Ching).
Specifically, they believed that the Way (Dao) of Heaven (Tian) is expressed in principle or li (理, py lǐ), but that it is sheathed in matter or qi (氣, py qì). In this, it is based on Buddhist systems of the time that divided things into principle (again, li), and shi (事, Pinyin shì). In the Neo-Confucian formulation, li in itself is pure and perfect, but with the addition of qi, base emotions and conflicts arise. Human nature is originally good, the Neo-Confucians argued (following Mencius), but not pure unless action is taken to purify it. The imperative is then to purify one's li.
Different Neo-Confucians had differing ideas for how to do so. Zhu Xi believed in gewu (格物, géwù), the Classification of Things, essentially a form of observational science, based on the idea that li lies within the world. Wang Yangming (Wang Shouren), probably the second most influential Neo-Confucian, came to another conclusion: namely, that if li is in all things, and li is in one's heart, there is no better place to seek than within oneself. His preferred method of doing so was jingzuo (靜坐, jìngzuò), 'quiet sitting', a practice that strongly resembles zuochan or Chan (Zen) meditation.
The importance of li in Neo-Confucianism gave the movement its name, literally "The study of Li."
Neo-Confucianism became the accepted state philosophy by the Ming, and continued in this way through the Qing dynasty and, in some ways, up to modernity. In China, the May Fourth movement, Communism and other modernizing movements have tried to wipe away the presence of Neo-Confucianism and met with a large degree of success. It continues to hold an influence in Mainland China and Taiwan, however.
Neo-Confucianism also gained acceptance in other areas. Many have argued that South Korea is the world's most Neo-Confucian state. See Yi Hwang and Yi I. Japanese Neo-Confucians such as Kaibara Ekken were instrumental in the formulation of Japan's dominant political philosophy.
In general, when people today discuss Confucianism, they often mean the Neo-Confucian formulation of it. The Confucian canon as it exists today was essentially chosen by Zhu Xi. Zhu codified the canon of Four Books (The Great Learning, The Analects of Confucius, Mencius, and Doctrine of the Mean) which in the subsequent Yuan and Ming Dynasties were made the core of the official curriculum for the civil service examinations.
List of Neo-Confucianists: