Two specific traditions attach to particular garnets, and they are worth separating from the generic red-stone lore.
In jyotisha, hessonite, the brownish orange grossular garnet, gomed, is one of the navaratna, and it is the stone of Rahu, the ascending lunar node. Rahu is not a body; it is a calculated point, the north node of the Moon's orbit, and in Indian tradition it is the severed head of the demon that swallows the Sun and Moon at eclipses. Its stone is accordingly regarded as powerful and unpredictable, and like blue sapphire it is prescribed with caution. That a mathematical point in celestial mechanics has a gemstone assigned to it is a good illustration of how far the jyotisha system extends its correspondences, and of the fact that it is a system rather than a collection.
In the Western Renaissance tradition, Agrippa's Behenian list pairs Regulus with garnet, and pairs Aldebaran with ruby or garnet as alternatives. Both are red stars in the older colour descriptions, and both get red stones; the reasoning is transparent and colour-driven.
Garnet is the January birthstone, and it held that position on the 1912 list and through every revision.
The medieval carbunculus, the glowing coal, a red stone said to shine in the dark, is usually taken to have meant garnet, sometimes ruby or spinel. No red stone glows in the dark. The claim is one of the durable fictions of the lapidary tradition, repeated from Pliny through the medieval encyclopaedists to the Renaissance, and it survived because the books copied each other rather than the stones.