Used by Eusebius of Caesarea, homoiousios means "of a similar substance". This is in contrast to the Nicene affirmation that Jesus and God the Father are homoousios, "of the same substance."
Christians at that time believed that even if they were of similar substance, the result was a Jesus who was not identical with the redemptive God of the Old Testament. Furthermore, if he had a similar divine substance, an immediate problem arises with the doctrine of monotheism. Thus, at the Council of Nicea the church affirmed that Jesus and the Father were of the same substance.
This is an improper term to use when describing the relationship between the Father and the Son. They are not of similar essence. Rather, they are of the same essence (Homoousios). In other words, according to the correct doctrine of the Trinity, the Father and the Son (and the Holy Spirit) share the same essence. See Ontological Trinity.