You can use a global variable, but that's not elegant.
A nicer approach is to use a static field: it is shared by all instances of your class, ie. only one field (and its value) exist for all objects of this type. Thus, it can hold a counter of the class instances:

1. void setup()
2. {
3.   Counted[] counted = new Counted[10];
4.   for (int i = 0; i < counted.length; i++)
5.   {
6.     counted[i] = new Counted("Name " + (i * 2 + 1));
7.   }
8.   for (int i = 0; i < counted.length; i++)
9.   {
10.     println(counted[i]);
11.   }
12. }
13. 
    
14. static class Counted // Have to be static go be a top-level class and accept a static field
15. {
16.   private static int count;
17.   private String name;
18.   private int id;
19.  
20.   public Counted(String _name)
21.   {
22.     name = _name;
23.     id = ++count; // Or count++ if you want to start IDs at 0
    
24.   }
25.  
26.   public String toString()
27.   {
28.     return name + ", ID: " + id;
29.   }
30. }

Of course, you can also create an ID in the loop creating the objects and pass it as parameter to the constructor (back to global variable if you don't create all the objects in the same loop).