The tiny crystals which formed around an exploding star are called stardust or presolar grains. Some of them will eventually form the rocks of the Earth. The atoms they contained are those able to condense at high temperatures, typically carbon (in the form of diamond or graphite) and silicon or aluminum combined with oxygen.
Interplanetary dust particle image from Jessberger
These dust grains are blown out of dying stars and mix with the original of the Galaxy to form dust clouds. The disc of the galaxy became thick with dust.
This dust is important for two reasons. One is that it provided the raw material for the formation of further stars and, perhaps even more important, for the planets which would form around them.
The second reason that stardust is important in this story is that it enabled the first molecules to form.
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