What is the specific task a singleton pattern accomplishes at the time it is handed .this?

Trying to get a handle on what the singleton pattern is, and what it's for. I think I'm close, but I'm not quite sure. I understand we use this technique when we only want one instance of a certain thing. But I haven't gotten my head around what that means and what the code part actually does.

FOR ANALOGY ONLY: Let's say I own a store with only one cash register. Employees and customers are scripts. I want every employee and customer in the building to use that one register for checkout related tasks. (I think that's what singleton is for)

 public class cashRegister
 {
     public static cashRegister TheeOneAndOnlyCashRegister;

     void Start()
     {
     }

     void Update()
     {        
     }

     void Awake()
     {
         if (TheeOneAndOnlyCashRegister == null)
             TheeOneAndOnlyCashRegister = this;
     }
 }

So what I don't understand is what I actually did here (if I even did this correctly).

Did I tell the store that there must be 'this' register in existence? Or did I tell the store that all people can only ever use this register? Or did I merely just name the register a name and demand the name was TheeOneAndOnlyCashRegister?