北京
Beijing is the name of the capital city of China. The first character, bei (北) means north and the second character, jing (京) means capital city. Beijing is how the Chinese term is written in hanyu pinyin, the official transliteration system used in China, Taiwan, and Singapore. Peking is the same Chinese word rendered in the romanization scheme of the Chinese Imperial Post Office. That romanization is fossilized in familiar terms such as 'Peking duck' (the food) and 'Peking man (the proto-human).
The correct pronunciation of the word Beijing in Hanyu (also referred to as the Mandarin language or putonghua) is 'bay' as in San Francisco Bay plus 'jing' as in Jingle Bells. This, of course, is ignoring the tones that are an essential part of the Chinese pronunciation. If you want to get it entirely right, pronounce the first part ('bay') with the third tone (low dipping-rising) and the second part ('jing') with the first tone (high level).
In the United States, one generally hears Beijing pronounced quite incorrectly with the 'j' of the 'jing' part sounding like the 's' in 'leisure' (i.e., /zh/), particularly by news announcers and others on TV, and by members of the government, who should really know better. It's not clear how this odd pronunciation came to be a de facto standard in US English (and apparently in other dialects as swell), as American English phonology provides perfect counterparts to the correct Chinese speech sounds (phones) and the 'j' represents exactly the sound we expect from an English 'j'.
Imperial Post Office Transliteration of Place Names