Divine appointments are those times when we most clearly perceive the hand of the Lord in arranging our circumstances, especially in relationship to other people.
For instance, when we're broken down on the side of the road and a kind person stops to help--who "just so happens" to be Christian--we reflect on the situation and consider how God put all that together. Or, those times when we're in a waiting room and notice a woman reading a book like The Case for Christ, so we strike up an evangelistic conversation and pray that she comes to know the Lord. These types of events are encouraging for us, as we sense God's involvement in our lives more deeply and tangibly.
But what about the 99.99% of life that isn't a "divine appointment"?