I get this search term a lot: "Registering a Small Business." People seem to think that they need to register something somewhere in order to conduct business. Well, yes and no.
You can start doing business under your own name by just by doing it. You are considered a sole proprietor running a sole proprietorship. No registration required anywhere to get that title.
If you conduct business under a name different from your own name, you need to register an assumed business name (aka fictitious business name or DBA for "doing business as") with the county clerk of the county where you conduct the business. You may have heard people say that you need to obtain a "business certificate" or "business license." Same thing.
If you decide to establish a more formal business entity to run your business, such as a a corporation or limited liability company, you need to file ("register") certain papers with the New York Department of State in order to form such an entity in accordance with the law. For example, to form a corporation, you need to file the corporation’s "certificate of incorporation."
Once you formed a corporation or LLC, (in most cases) you need to obtain ("register") a federal tax identification number (aka employer identification number) in order to open a bank account in the name of your corporation or LLC. Find out more here.
If you formed a corporation, you may want to elect a special tax status for your corporation in order to become an S corporation by filing ("registering") Form 2553 with the Internal Revenue Service. You have to do a similar election with the New York tax authorities.
If you engage in certain business activities, you may need to apply (again, you may think of "register") for permits and licenses; for example, a liquor license for a restaurant. To find guidance with respect to the licenses and permits you need for your New York business, check New York State’s Online Permit Assistance and Licensing website.
This is no legal requirement, but you really should register for your own domain name in order to give your business the web presence it deserves.
Finally, if you plan to develop a brand identity, you might consider protecting your business name and/or name for your product or service by "registering" a trademark.