We chant this hymn during Vespers on the feast of Saint Eleutherios on December 15th. The name Eleutherios means "one who is free" or "one who gives freedom", and is therefore the patron saint of those who seek to be free of something, whether it be free from prison, free from oppression, or free from disease.
In northern Greece especially, the most popular wish bestowed on a pregnant woman is "Kali Leuteria", which means "Good Freedom", especially towards the end of a pregnancy. A pregnant woman will also pray: "Saint Eleutherios, free me" (Αϊ-Λευτέρη, λευτέρωσέ με). This is because giving birth could carry a variety of complications, therefore when you wish someone "Kali Leuteria" you are wishing that they have an easy and harmless childbirth. The prayer is that the mother would be given her freedom in the best possible way.
Sometimes in Greece, when a mother is having a particularly hard time giving birth, she will pray to Saint Eleutherios to free her, and in return when her prayer is granted she will return the favor and name her child Eleutherios if it is a boy or Eleutheria if it is a girl.
On the island of Lesvos, midwives also celebrate on this day, and a prayer is said that the births they help deliver will go well.