Pottery is dishes, plates, cups, and cooking pots made out of clay. It is a good idea to make dishes and pots out of clay for several reasons. Clay is cheap and easy to get, pretty much anybody can make a useful pot out of it, and you can make it waterproof pretty easily too. It's relatively easy to clean. Plus it can be made very beautiful, if you know what you are doing. And it is easy to make yours look different from your neighbor's.
People first started making pottery out of clay in East Asia, in both China and Japan, around 14,000 BC, long before they started farming. Probably they had always known how, but just hadn't done it much. This early pottery was made by just pushing a hole into a ball of clay, or by making a long snake of clay and coiling it up into a pot shape. It may have gotten started by making baskets and coating them with clay. In Japan, early pots might be buried in the ground for storage. One reason for starting to make pottery in Japan may have been to preserve fish by fermenting it into fish sauce in these buried pots.
People probably began to make pottery in the Americas for similar reasons, though several thousand years later. People who ate a lot of fish and shellfish were making pottery in Brazil about 5500 BC, and maybe they also used pottery jars to preserve fish by fermenting it. From Brazil, people gradually began using pottery further north, perhaps spreading to fish-eating ancestors of the Cherokee and other Mississippians in what is now Florida and Georgia by about 4500 BC, and then to the west side of South America.
About the same time, the use of pottery spread west from East Asia, reaching Mesopotamia and the Eastern Mediterranean, and then North Africa, around 6000 BC, near the beginning of the Neolithic period there. West Asian and African people may have begun to make pottery as a way of storing grain safely when they started farming. But, like people in Japan, they also used pottery to make fermented fish sauce.
By around 3000 BC, at the beginning of the Bronze Age, people in West Asia had begun to use the slow potter's wheel. This is a little platform made of wood that you build the pot on; you can turn it around so that instead of having to walk around your pot you can sit still and turn the pot around. In the hands of someone who is good at using it, it does make potting a lot faster.
A woman in Mexico digging clay and using a slow wheel to make a pot.
The slow wheel was also invented in Central America, where the Zapotec were using it to make pottery, maybe by around 100 BC. The Zapotec kept right on using the slow wheel, but by 2000 BC, the slow wheel had been almost entirely replaced in Europe and Asia by the fast wheel, which is also a platform, but one which spins on an axle, like a top. You can start it spinning with a push or a kick, and then draw the pot gradually out of the lump of clay. Using the fast wheel, a good potter can make a pot every minute or so, and all of them almost exactly the same. It's much faster than coiling or the slow wheel, and so pots got much cheaper than they had been before. The Indo-Europeans, migrating at this time into Greece and Italy and China, brought the idea of the fast wheel with them to all of those places.
From the beginning, people used pottery as a way of constructing their social identity, or showing who they were and how they were different from other people. Many of the designs used on pottery were borrowed from cloth, which was also used to identify people of one group or another. Greek pottery is very different from West Asian pottery of the same time, and both of them are different from Egyptian pottery, or Chinese pottery. Etruscan pottery is different too, but similar to Greek pottery in many ways.
By this time, pottery was spreading to almost everywhere in the world. People started using pottery in the Pueblo region of the American south-west about 1500 BC, and along the Mississippi valley and the Blackfoot about 200 BC, and in West Africa about 400 BC.
The beginning of the Roman Empire saw some big technological and economic changes in the Western pottery industry. First, people began painting pottery red instead of black. Then they began making it in molds instead of painting it. Around the same time, the Phoenicians invented glass-blowing, and this made glass cheap enough to be a serious competitor with pottery. People pretty much stopped making pottery cups, and everyone drank out of glasses. Even a lot of bowls, and little things like perfume containers, were made out of glass.
The Arab invasion of North Africa around 700 AD ended the North African pottery trade, and after that pottery was locally made again for some time in the West, and not very good. Although the use of pottery was still spreading, reaching Congo in Central Africa by around 900 AD, there were still parts of the world where people relied entirely on baskets: South Africa, Australia, and the Chinook people of North America.
The next great developments in pottery were not in the West but in Sui Dynasty China, where potters began to make porcelain (PORR-se-lenn) cups and pitchers around 700 AD. This gleaming white pottery was popular not only in China but in West Asia too. But it was very expensive in West Asia, because it had to be carried all the way from China on donkeys and camels. So the West Asian potters invented lead glazes, which made ordinary pots look white and shiny. This made a kind of imitation porcelain which was a lot cheaper.
A little later on, European and Chinese potters began using lead glazes too. About 1200 AD, potters of the Yuan dynasty in China began to use different color glazes to create designs on their pots. Chinese pottery was still the best and the most expensive. So West Asian potters also used these colored glazes to imitate Chinese designs, and Europeans used colored glazes to imitate the West Asian designs.
Bibliography and further reading about the history of pottery:
The Kids 'N' Clay Ceramics Book, by Kevin Nierman (2000). For kids who want to throw their own pots. Pretty ambitious.
Fired Up!: Making Pottery in Ancient Times, by Rivka Gonen (1993). For kids. Watch out - there are some inaccuracies. Still, it's the best available history of pottery I can find for kids.
Ten Thousand Years of Pottery, by Emmanuel Cooper (2000). Not for kids, but a better account. Continues up to modern times.