What happened between the time of the Old Testament and the New Testament?

Lots of things. Many people refer to this period as the Intertestamental Period. Many important developments for the understanding of the NT took place; for instance, the group referred to as Pharisees took shape. Many of the happenings of this period are recounted in the books called Deutero-Canonical by the Roman Catholic Church and Apocryphal by Protestants. For more information, one might read these books, which include the likes of 1 and 2 Maccabees. Or, one might consider a course of study on the History of Israel.