Great Burgher refers to the burgher right. There was also a small burgher right. In order to do business in New Amsterdam, one had to purchase either the Great or the Small right to do so. Burghers swore an oath of loyalty to the government and to the community, the burgher oath. So being a Great Burgher conferred both business opportunities, and increased status.
Please see Jaap Jacob's detailed description of the evolution of the Burgher Rights in New Amsterdam here:
https://books.google.ch/books?id=vgnh3E5Mm0cC&pg=PA192&lpg=PA192&dq=great+burgher+right+new+amsterdam&source=bl&ots=mj2gacez-2&sig=qKVVBxsgMta9lXCzQBp0MG_3T80&hl=en&sa=X&ved=0ahUKEwjr0pmRg5XUAhUEIlAKHZQ9CKAQ6AEIMzAC#v=onepage&q=great%20burgher%20right%20new%20amsterdam&f=false