Let me quote my own answer to a related question, Difference in pronunciation between “щ” and “шь”?:
щ, it is a long alveopalatal consonant, IPA symbol for it is /ɕː/ (you can listen to the sound there, and there is also the sound for the Russian word 'счастье' [ˈɕːæsʲtʲjə] in which 'сч' is pronounced as 'щ'). Roughly speaking, in Standard Russian it is pronounced like the English 'shsh' in the phrase 'the dish she likes'. Some people pronounce
щ as 'шч', but this pronunciation is considered obsolete.
The fact is, that although now the standard pronunciation of
щ is /ɕː/, it used to be pronounced as /ɕtɕ/ and is still pronounced so by many people, that is why it is traditionally transliterated as
shch in English. Note, that when the Cyrillic script emerged in the 9th century,
щ stood for the consonantal cluster [ɕtʲ], and later in Russian [ɕtʲ] > [ɕtɕ] > [ɕː]. Thus, the tranliteration of this letter as
shch reflects the previous stage of its pronunciation which is dying out now.