Eight is two cubed, the first cube after one, and the number of regeneration, cycles and the double balance, the figure of two squares or of the octagon, the shape that mediates between the square of earth and the circle of heaven in much sacred architecture, which is why baptismal fonts and baptisteries were so often eight-sided, baptism being the new birth into the eighth day. Eight recurs as the number of renewal beyond a sevenfold completion: the eighth day, the octave that returns the note an octave higher, the resurrection on the day after the sabbath. Turned on its side it is the sign of infinity, and in Chinese culture eight is the most auspicious number of all, its name close to the word for prosperity. From this come its associations of cycles, regeneration, and material and cosmic balance.
In modern numerology eight is the number of power, ambition, money, authority and worldly achievement, the executive, the number of material mastery. That reading is a twentieth-century American construction rather than an ancient doctrine, though the associations of eight with regeneration and the octave are genuinely old.