Private functions are functions within the classes that stays private, in other words no functions outside of that class can access that function.
Public is the opposite. Let's say you have class A with function AA and class B with function BB. If AA is public, then BB can access that function for whatever reason that may be, but if AA is set to private (private function $AA()) then BB can't access it.
I've never dealt with PHP OOP, only ActionScript, but I'm guessing the constructor is only a function that is called to construct that object.