The acronym 'GmbH', which is written after the name of a German company, designates it as a private entity. The letters stand for Gesellschaft mit beschränkter Haftung which, translated literally, means a 'company with limited liability'. GmbH companies are incorporated and, as such, are legal entities unto themselves. They can be owned by individuals, public companies, or any other sort of legal entity.
German companies that are publicly traded are designated as such by the letters 'AG' after the company name. 'AG' is an abbreviation for the German word Aktiengesellschaft, which literally translates to 'stock corporation' or 'shares corporation' in English.
The laws of a country determine which types of companies are legally recognized within the country's borders. One of the most commonly used acronyms is 'PLC', which is used throughout the United Kingdom to indicate a Public Limited Company, or a company that is publicly traded with owners having limited liability. Throughout the world, 'S.A.' is used to indicate a privately held company, though its exact meaning varies depending on the country in which the company is registered. The words translate broadly as 'anonymous society' in English.
More information on company extensions can be found here.
For related reading, see The Requirements for Being a Public Limited Company and The Basics of Forming a Limited Liability Company (LLC).