Securing a business name
There are several ways to secure your business name. Let me describe the major ones:
1) Registering a DBA: if you choose not to incorporate or organize your business as LLC you can register your business name as "doing business as" or DBA. In most states the DBA is registered with the county, so this gives you a somewhat limited right to your business name.
2) Incorporating or Forming an LLC: when you incorporate or form an LLC in any given state you would be able to reserve your business name in that state. You can use this business name everywhere, however only reserves it for your exclusive use in the state of formation only (or any other state where you will register your company).
3) Registering a Trade Mark with USPTO: you can register the business name as a Trade Mark with US Patent & Trademark Office, which will reserve the name for your exclusive use in all 50 states and DC. You should use the help of an IP attorney, and as a result this solution is often on the expensive side.