To become an officially recognized business entity, you must register with the government. Corporations will need an "articles of incorporation" document, which includes your business name, business purpose, corporate structure, stock details and other information about your company. Otherwise, you will just need to register your business name, which can be your legal name, a fictitious "doing business as" name (if you are the sole proprietor), or the name you've come up with for your company. You may also want to take steps to trademark your business name for extra legal protection.
LLC, LC, Ltd. Co., Limited Liability Company: a form of business whose owners enjoy limited liability, but which is not a corporation. Allowable abbreviations vary by state. Note that in some states Ltd. by itself is not a valid abbreviation for an LLC, because in some states (e.g. Texas), it may denote a corporation instead. See also Series LLC. For U.S. federal tax purposes, in general, an LLC with two or more members is treated as a partnership, and an LLC with one member is treated as a sole proprietorship.
LLC members are not personally responsible for the company's debt or liability. S-Corp shareholders are not personally responsible for the company's debt or liability. C-Corp shareholders are not personally responsible for the company's debt or liability. Non-Profit directors are not personally responsible for organizational debt or liability. Sole Proprietors are personally responsible for debt and liability.
The "mit beschränkter Haftung (mbH)" suffix (German: [ˈɛmbeːˌhaː], "with limited liability") is sometimes added to the name of a firm that already ends in "-gesellschaft" ("company"), e.g., "Mustermann Dental-Handelsgesellschaft mit beschränkter Haftung" ("dental trading company with limited liability"), which would be abbreviated as "Mustermann Dental-Handelsgesellschaft mbH".