Alabaster is a translucent stone, usually white, that is used for sculpture and interior decorations. It consists of the mineral gypsum with a very fine grain, massive habit, and even coloring.
"Alabaster" is also used to refer to a similar type of marble, but a better name for that is onyx marble, or even better just marble. Onyx is a much harder stone, composed of chalcedony, with straight bands of color instead of the curved forms typical of agate. So if true onyx is a banded chalcedony, a marble with the same appearance should be called banded marble instead of onyx marble, and certainly not alabaster, which is not banded at all.
There is some confusion because the ancients used gypsum rock, processed gypsum and marble for the same purposes under the name alabaster.