Around 530 BC, Athenian potters were more and more frustrated by the black-figure way of vase-painting. They wanted to paint figures that overlapped, for instance, which was very difficult to do in black figure without the whole thing looking like just a big black blob. And they wanted to be able to show the muscles better too.
So somebody had an idea: instead of painting the people black, why not paint the background black and leave the people red? This is harder because you have to carefully paint all around the people in the picture, but it makes the people look much more real. The slip and the firing are exactly the same as in black figure.
Some of the greatest Greek vases are in red figure. One of the most famous painters is the Berlin Painter.
But by around 450 BC, just eighty years after the invention of red-figure painting, hardly any potters were still making red-figure vases. Nobody really knows why they stopped. Maybe red-figure just went out of style. Maybe the Athenians became so rich that they all used metal (bronze or silver) dishes instead of pottery.
It's more likely that the Athenians were rich enough from their empire that they didn't need to sell their pottery to other people. Most Athenian pottery had always been produced to sell in other countries. Also, the Etruscans, who had bought a lot of the Athenian pottery, were no longer doing very well by 450 BC, and maybe they couldn't afford to buy Athenian pottery anymore.
One kind of pottery which does last longer is the white-painted lekythos, which was placed on graves, like a tombstone. Potters kept on making these pottery tombstones until about 400 BC. Then even these stopped. After 400 BC, Athenian pottery - and all Greek pottery - is just plain, without pictures.
Athenian Red Figure Vases: The Archaic Period : A Handbook, by John Boardman (1985)
Athenian Red Figure Vases: The Classical Period : A Handbook, by John Boardman (1989)
The Archaeology of Greece: An Introduction, by William R. Biers (1996)
Greek Art and Archaeology (3rd Edition), by John G. Pedley (2002).