Mandarin is national language of China, Taiwan, and some other countries such as Singapore. In a huge country like China, people from different region speak different dialect. Thus, to unity the people mandarin is used. Commonly for official papers, etc.
About your question, traditional or simplified Chinese, they're the writing method. Traditional Chinese is the original character, meanwhile simplified Chinese is the simplified method (the word is self-explained). To make it simple, I'll show you the difference. The word "please" is written 請 in traditional Chinese and 请 in simplified Chinese (both are pronounced qĭng). Now, you can see that traditional Chinese has more strokes than simplified Chinese.
Traditional Chinese is used in Hong Kong and Taiwan, simplified Chinese in China and Singapore.