Changing the name of the company
You really have two choices there. First, you can change the name of your company completely. That's done by filing an amendment to the Articles of Incorporation (for corporations) or Articles of Organization (for LLCs). This way your company will no longer be known by its original name, but will now operate under its new name.
Your second choice is to apply for a fictitious name or a DBA. This will allow your company to do business under the new fictitious name, but also be known by its original name.
Of course, in either case you first have to make sure that your new name is available.