Constructors or Methods

When you do Order o = new Order(1);
It means you are creating a new memory space for object o of order type.
When you do Order o = OrderManager.GetOrderById(1);
It means that you are assigning reference to object o of type order which is return by GetOrderById fuction of class OrderManager.
This is not any rule of C# but it is your own way(logic) to retrive the value.

Well, the first line is creating a new Order object using the Order constructor.

The second line is returning a reference to an existing Order object by calling the method OrderManager.GetOrderById and passing it an argument of 1.

I would imagine that the Order objects are stored within the OrderManager class and have an Id property to uniquely identify them which is passed in the constructor.

So, it's not so much a matter of style or rule - it's a question of creating a new object or accessing an existing one that is stored in some collection class.

In C#, constructors always return a reference to a new object of the class in question.

Other methods may or may not return references to new or existing objects - it depends what they're designed to do.