Write a custom Roll-Up Summary trigger to count the total number of Contacts on an Account (handling Insert, Update, Delete, Undelete, and Account Reparenting).

trigger ContactRollup on Contact (after insert, after update, after delete, after undelete) {
    Set<Id> accIds = new Set<Id>();
    List<Contact> src = Trigger.isDelete ? Trigger.old : Trigger.new;
    for (Contact c : src) {
        if (c.AccountId != null) accIds.add(c.AccountId);
        if (Trigger.isUpdate) {
            Id oldId = Trigger.oldMap.get(c.Id).AccountId;
            if (oldId != null) accIds.add(oldId);
        }
    }
    Map<Id, Integer> counts = new Map<Id, Integer>();
    for (AggregateResult ar : [SELECT AccountId a, COUNT(Id) c FROM Contact
                                WHERE AccountId IN :accIds GROUP BY AccountId]) {
        counts.put((Id) ar.get('a'), (Integer) ar.get('c'));
    }
    List<Account> toUpdate = new List<Account>();
    for (Id id : accIds) {
        toUpdate.add(new Account(Id = id, Contact_Count__c = counts.containsKey(id) ? counts.get(id) : 0));
    }
    update toUpdate;
}