You can usually choose any name you want as long as it is not already used or is not considered obscene or objectionable by your local Corporate Filing Office. When articles of incorporation are filed, the Corporate Filing Office will check its database and tell you if the name is already being used. If so, you will have to make a change. Also, corporate laws generally require that your chosen name end in some word or abbreviation that denotes a corporation. These include incorporation (or inc.), corporation (or corp.), company (or co.), or limited (or ltd.)
Your corporate name will only be registered in the state you incorporate in or in which you apply to a foreign corporation. There may be other corporations with the same name in other states. Most states now provide online searches to the public for a corporate name. Click the link for a list of Corporate Filing Office websites by state.