Hi Simon,
I think I can answer your question. Some years ago I bought one of those $50 reference manuals on C# and basically what it says is that a "field" tells us what the variables of a Class are, and the "properties" tell us how those fields work.
Part of the problem is that, when using Visual Studio, we don't actually see what the underlying C# code is doing ... Visual Studio handles a lot of that for us.
For instance, using your examples above, you might often see Bob type:

   prop [tab] [tab]

to generate the code for properties, and get something like your example:

   public int Health {get; set;}

This is actually a shorthand that VS can understand. The actual underlying code would be:

public class Game
{
	private int Health;	//defines the field "Health"


	public Game(int health)	// a constuctor using one property
	{
	 this.Health = health;	 // matches the field to the property
	}

	public int Health	 // the code that makes Health work
	{
	    get
	    {
	       return health;
	    }
	    set
	    {
	        health = value;
	    }
    }
}

Hope this helps

 - john