An aragonite sea contains aragonite and high-magnesium calcite as the primary inorganic calcium carbonate precipitates. The chemical conditions of the seawater must be notably high in magnesium  content relative to calcium (high Mg/Ca ratio) for an aragonite sea to form. This is in contrast to a calcite sea in which seawater low in magnesium content relative to calcium (low Mg/Ca ratio) favors the formation of low-magnesium calcite as the primary inorganic marine calcium carbonate precipitate.
The Early Paleozoic and the Middle to Late Mesozoic oceans were predominantly calcite seas, whereas the Middle Paleozoic through the Early Mesozoic and the Cenozoic (including today) are characterized by aragonite seas.
Aragonite seas occur due to several factors, the most obvious of these is a high seawater Mg/Ca ratio (Mg/Ca > 2), which occurs during intervals of slow seafloor spreading. However, the sea level, temperature, and calcium carbonate saturation state of the surrounding system also determine which polymorph of calcium carbonate (aragonite, low-magnesium calcite, high-magnesium calcite) will form.
Likewise, the occurrence of calcite seas is controlled by the same suite of factors controlling aragonite seas, with the most obvious being a low seawater Mg/Ca ratio (Mg/Ca < 2), which occurs during intervals of rapid seafloor spreading.