What does "seraphic" mean?
The word "seraphic" is a reference to the seraphim, a special class of angels. According to the Catholic Encyclopedia, "The name is oftentimes derived from the Hebrew verb saraph ('to consume with fire'), and this etymology is very probable because of its accordance with Isaiah 6:6, where one of the seraphim is represented as carrying celestial fire from the altar to purify the Prophet's lips."
Tradition states that it was a seraph that gave to St. Francis the wounds of Christ on his body (commonly known as the stigmata).
While there does not exist a similar miracle for St. Clare, she shared with St. Francis a deep and passionate love of God, which is a characteristic of the seraphim.
With this in mind, many Franciscans still call Sts. Francis and Clare "seraphic," honoring the extraordinary example of holiness they gave to the world.