There is no specific form required by NY state to show an affiliation with a particular brokerage. NY state does give your sponsoring broker your actual license to hold during the time you are associated there. You will receive a pocket license to keep with you as you may from time to time be required to show it for access to listings, etc.
If you change brokerages, you will be required to pay a fee that disassociates you from one firm and associates you to another. When you've completed enough deal to equal 1700 points, you can then take the NY state required course of study to obtain your Brokers license. After that, you can either open your own R.E. business or remain with an established brokerage. You would then be considered a Broker Associate instead of just a licensed Agent.