CLAY POT COOKING
There are essentially 3 types of clay bodies used for cooking in clay; Stoneware (most Common), earthenware, and flameware.
STONEWARE:
Stoneware is typically a variety of clays and other ingredients that are blended for firing temperature, throwing or hand-building characteristics, fired color and impact on glazes.  Stoneware is defined as pottery fired to at least 2000 degF.  which makes the clay essentially non-porous and very hard. In the firing of stoneware, there is always some residual crystalline silica.  These silica crystals expand and contract when heated and cooled.  It is this expansion that can cause stoneware to crack when exposed to heat.  Particularly troublesome is a crystal called cristobalite which very rapidly expands about 3% at 437 degF.    Secondarily, if the stoneware pot is subjected to a very rapid heating or cooling, like taking a pot out of a hot oven and placing it on a cold surface, the result can be a crack due to different expansion rates in different parts of the pot. This is what will cause most failures.
These can be avoided by putting the cold pot in a cool oven and bringing both up to temperature together.  Similarly a cold roast can cause a cold spot which can cause a crack.  This can be avoided by putting a small amount of water in the pot to spread the stress.
Clay bodies can also be formulated to melt the silica crystals into the body, avoiding most of the expansion. This can be tested for using an instrument called a dilatometer.
Which all goes to say, if you avoid direct heat (stovetop), and are generally careful to avoid rapid temperature change, you should have no problem with stoneware cooking pots for oven, microwave and dishwasher.
CHINA is really a subset of stoneware, fired to a vitreous or near vitreous state and often times glazed over its whole surface. Because it is white, glazes maintain their color. The highest fired china is porcelain.  For the most part, china has the same expansion issues as stoneware.

EARTHENWARE
Lower fired clays (1800 to 2000 degF) are considered earthenware.  They may be glazed or unglazed (think Romertopf) and some of them, with care, can even take direct heat.  Because of the lower firing temperature, earthenware is typically still porous but crystalline silica is not formed.  It is somewhat fragile because of the lack of vitrification of the clay body.

More to come…see also flameware on this site.