The name "Peking" is simply a western conventional spelling, accepted internationally for a very long time. Under the Wade-Giles romanisation system, the name was rendered Pei-ching, which means "northern capital". In 1928 the nationalist government moved the capital to Nanking and changed the name of the old city to Pei-p'ing or Peiping ("northern peace"). Thus it remained until 1949, when the Communists took over and restored the old capital, changing its name back to Pei-ching.
In 1958, the Chinese government introduced a new and official system of romanisation called Pin-yin (phonetic spelling). Under the Pin-yin system, the name of the city of Peking is rendered Bei-jing, which is now the official Chinese spelling using Roman characters. The actual name did not change.
The nationalists of Taiwan have never accepted Pin-yin, and still refer to Peking. I don't think the Chinese as a whole mind what we call the city.
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