How do I delete an item or object from an array using ng-click?

To remove item you need to remove it from array and can pass bday item to your remove function in markup. Then in controller look up the index of item and remove from array

<a class="btn" ng-click="remove(item)">Delete</a>

Then in controller:

$scope.remove = function(item) { 
  var index = $scope.bdays.indexOf(item);
  $scope.bdays.splice(index, 1);     
}

Angular will automatically detect the change to the bdays array and do the update of ng-repeat

DEMO: http://plnkr.co/edit/ZdShIA?p=preview

EDIT: If doing live updates with server would use a service you create using $resource to manage the array updates at same time it updates server