Ancient Greek pottery, the pottery of Ancient Greeks, is important to us firstly for the intrinsic beauty of its forms and decoration and secondly, for the light it sheds on the development of Greek pictorial art.
Because fired clay pottery is highly durable —and few or no Greek works in wood, textile, or wall painting have survived— the painted decoration of this pottery has become the main source of information about the process whereby Greek artists gradually solved the many problems of representing three-dimensional objects and figures on a flat or curved surface.

The large number of surviving examples is also the result of a much wider reliance on pottery vessels in a period when other materials were expensive or unknown.
Ancient Greek pottery vessels was used by the Greeks primarily to store, transport, and drink such liquids as wine and water. Smaller pots were used as containers for perfumes and unguents.
Manufacture of Ancient Greek Pottery
Ancient Greek pottery techniques
Construction

Clay powder
The process of making a pot and firing it is fairly simple. The first thing a potter needs is clay. When clay is first dug out of the ground it is full of rocks and shells and other useless items that need to be removed. To do this the potter mixes the clay with water and lets all the impurities sink to the bottom. This is called levigation or elutriation. This process can be done many times. The more times this is done, the smoother clay becomes.

A potter's wheel
The clay is then kneaded by the potter and placed on a wheel. Once the clay is on the wheel the potter can shape it into any of the many shapes shown below or anything else he desires.
Ancient Greek pottery pots were usually made in sections such as the body and feet and spout. Even the body, if it were larger than 20 centimeters, might be made in separate sections and glued together later with a thin watery clay called slip. After the pot is made then the potter paints it with a very pure black slip (made from the same clay) and a brush.
Ancient Greek pottery, unlike today's pottery, was only fired once, but that firing had three stages. After the pottery is stacked inside the kiln our potter can start the first stage. He heats the kiln up to around 800°C with all the vents on the sides open to let air in. This turns the pottery and the paint red all over. Once the kiln reaches 800°C the vents are closed and the temperature is raised to 950°C and then allowed to drop back to 900°C. This turns the pottery and the paint all black.
The potter then starts the third and final phase by opening the vents and allowing the kiln to cool all the way down. This last phase leaves the slip black but turns the pottery back to red. This happens because when the clay is given air it turns red, but when the black slip is heated to 950°C it no longer allows air in. So the slipped area stays black while the bare areas stay red.
Wheel made pottery dates back to roughly 2500 BC where before the coil method of building the walls of the pot was employed. Most Greek vases were wheelmade, though as with the Rhyton mould-made pieces (so-called "plastic" pieces) are also found and decorative elements either hand formed or by mould were added to thrown pots.
More complex pieces were made in parts then assembled when it was leather hard by means of joining with a slip, where the potter returned to the wheel for the final shaping, or turning. It was then slipped and incised ready for the kiln.
Decoration and firing of Ancient Greek pottery
The striking black slip with a metallic sheen, so characteristic of Greek pottery was a fine suspension (colloidal fraction) of an illitic clay with very low calcium oxide content which was rich in iron oxides and hydroxides, differentiating from that used for the body of the vase in terms of the calcium content, the exact mineral composition and the particle size. This clay suspension was most probably collected in situ from specially located illitic clay beds that produced spontaneous colloidal dispersion in rain water.

Gas operated pottery kiln
The stability of the chemical composition of the Attic black slip argues against the use of added deffloculants such as wood or other plant ashes, urea, tannins, even blood, suggested by several authors during the 20th century. This clay suspension was thickened by concentration to a paste and was used for the decoration of the surface of the vase. The paint was applied on the areas intended to become black after firing.
The black color effect was achieved by means of changing the amount of oxygen present during firing. This was done in a single cycle, in a process known as three-phase firing.
First, the kiln was heated to around 920-950°C, with all vents open bringing oxygen into the firing chamber and turning both pot and slip a reddish-brown (oxidising conditions) due to the formation of hematite (Fe2O3) in both the paint and the clay body. Then the vent was closed and green wood introduced, creating carbon monoxide which turns the red hematite to black magnetite(Fe3O4); at this stage the temperature decreases due to incomplete combustion.
In a final reoxidizing phase (at about 800-850 °C) the kiln was opened and oxygen reintroduced causing the unslipped reserved clay to go back to orange-red. In the previous phase, chemical composition of the slipped surface had been altered, so it could no longer be oxidized and remained black.
An excellent video from the Getty will show you the making of a greek pottery.