English people say Peking and Beijing. The Chinese government wants people to say "Beijing" because "北京" is pronounced Beijing in Mandarin Chinese.
English and American people use both Peking and Beijing, although in written texts Beijing is used more and more.
Beijing and Peking are both correct romanizations of the same word. Beijing is the pinyin romanization, and Peking is the Chinese postal romanization of the same 北京.
Every language based on Roman script have both spellings, but both PRC and ROC adopted pinyin (and Beijing) as their official romanization. The adoption of pinyin in other countries vary.
Most of the English-speaking countries adopted Beijing as a preferred name of the Chinese capital, when talking about current affairs. However, even Chinese official names of Peking University, Peking Opera etc use the old postal system. When referring to the past, many English sources also continue to use postal romanization: Richard Nixon visited Peking and met Mao Tse-tung.
Hungary, however, have not adopted pinyin even on governmental level: the Hungarian embassy uses "Peking". (Incidentally, German embassy and other governmental sources use Beijing, while German newspapers continue with Peking).
So, when translating from Hungarian to English, both Beijing and Peking are valid. But in English->Hungarian translations Peking is the only choice.