This month's Illinois Bar Journal has a very good article on recent developments in grandparent visitation law. Below is a really short explanation for non-lawyers:
You have probably heard that child custody decisions are made with the "best interests of the child" in mind. This rule generally only applies for parents. For example, the court will use the "best interest of the child" standard when deciding how much visitation a parent should have.
However, if you are a grandparent, you cannot rely on the "best interests of the child" to force a reluctant parent to grant you visitation. A grandparent must "produce evidence which demostrates to a judge that a strong emotional bond has developed between the grandparent and the child and that the health, safety and welfare of the child will be adversely affected if visitation is entirely cut off." ISBA Law Journal (May 2008).