Fear of abandonment is one of a child's first responses to divorce; they reason that if one parent can leave, then obviously the other can as well. Consequently, children need constant reassurance that they will not be abandoned. Ideally, this will come from both parents.
Another common worry of children is that the parent moving out will not have a place to live, not be taken care of, etc. One way of possibly eliminating or preventing this reaction is to, as soon as possible, showing the child the new home of the departing spouse.
Other reactions that are quite common include: sleep disturbances (nightmares, inability to sleep, etc), fears of impending disaster, suspiciousness, under-achievement in school, faltering peer relationships, emotional constriction, anger, and regression in behavior. Children are likely to feel vulnerable and powerless both during and following a divorce; when children are deeply attached to both parents, they often wish to stop the divorce. They may blame themselves, suspecting that if they have behaved differently, the divorce would not have happened. Consequently, they need to be reassured, repeatedly, that they are not to blame, that they did nothing wrong, etc.
Obviously, the manner in which children are informed about the divorce is important and frames their conception of it. Consequently, this should be done with care and sensitivity. However, it is not advisable to tell the child that the divorce does not matter, as it obviously does. Rather, one should attempt to provide an honest explanation that does not dwell on issues that are beyond the child or that the child does not need to hear (specific details of infidelity, etc). The child should be told what is different and what will remain the same. Finally, children should be given the opportunity, on many occasions, to express their opinions and to ask questions. They should know that things will be different, but will improve with time. So that children can deal with divorce in the best manner possible, it is best that they remain close with both parents.