(Simplified Chinese: 普通话; Traditional Chinese: 普通話; literally "common speech").
Pǔtōnghuà, often referred to in English as "Mandarin Chinese", "Standard Mandarin" or simply "Mandarin", is the official language of the People's Republic of China and is understood by the majority of Chinese speakers.
In Taiwan, a more common term for pǔtōnghuà is guóyǔ (Traditional Chinese: 國語; Simplified Chinese: 国语; literally "national language"). It is also known as hànyǔ (Simplified Chinese: 汉语; Traditional Chinese: 漢語; literally: "Han (Chinese) language").
Pǔtōnghuà is based on a standardised form of the language used around Beijing, although there are differences between the Beijing dialect pronunciation and Standard pronunciation.
English-speakers referring to "Chinese" as a language will probably mean Mandarin. However, they may simply mean what they most frequently hear Chinese people speaking, which could be any one of a number of dialects such as: Cantonese, Shanghainese, etc.
Hànyǔ Pīnyīn is the official romanisation of pǔtōnghuà in the People's Republic of China. Taiwan typically makes use of the Wade-Giles system.