If one person requires an apology before forgiving and another person practices unconditional forgiving, this does not necessarily imply a large difference in their understanding of forgiveness. For Jewish and Christian people, forgiveness is an act of mercy toward a person or people who have acted unjustly toward the forgiver. In both monotheistic traditions, people see that all persons are made in the image and likeness of God. This insight makes forgiveness appropriate because even those who behave badly are made in that image.
Please keep in mind that some in the Jewish tradition practice unconditional forgiveness, as Joseph did when forgiving his brothers in the book of Genesis. Some Christians require an apology before they forgive. In terms of the essence of what forgiveness is, however, people from both traditions tend to share the understanding that to forgive is to practice love and mercy toward the wrongdoer.