The English word "china" comes from the country: China invented porcelain, and English simply borrowed the country name for the ware.
A thousand years of kilns: Jingdezhen in Jiangxi is the "porcelain capital", from Song qingbai ware to Ming-Qing imperial kilns; its porcelain circled the globe, traded on the Silk Road for a millennium.
The great families: blue-and-white (qinghua), famille-rose (fencai), Ru ware (sky-after-rain celadon, first of the five great kilns, fewer than a hundred pieces survive), Jun ware (kiln-transformed colours), Longquan celadon.
Why it is precious: fine ware passes many steps - throwing, glazing, firing at 1,300 degrees; the Ru "sky-clearing-after-rain" glaze is legendary and extremely rare.
Where to see/buy: Jingdezhen Ancient Kiln Folk Museum and the China Ceramics Museum; every museum has a porcelain hall; for souvenirs look for "Jingdezhen made" - street "antiques" are fakes.
Cultural life: porcelain carries the tea and food cultures - a Yixing teapot or a set of blue-and-white tableware is a proper Chinese way to host.