Russian ‘щ‘ can sound a whole lot like ‘ш‘ to the untrained, newbie ear! It is important to pronounce them both correctly, because they are both unique sounds!
Some resources will tell you that ‘щ‘ is pronounced as “shch”, as in “fresh cheese.” This is not correct in modern Standard Russian! Although Russian words with ‘щ‘ are transcribed as “shch” or “sch” (e.g. борщ = borsch), the pronounciation is [ɕː] nowadays. It is basically a long, palatalized version of English’s “sh” as in “ship.”
In the Polish language, this sound is still “shch”, transcribed as [ʂt͡ʂ] in IPA and written as ‘szcz.’
‘щ‘ is a voiceless alveolo-palatal fricative. It is a postalveolar fricative (like the ‘sh’ in English “ship”), but it is palatalized, meaning that the middle of the tongue is raised towards the hard palate (thus it is a soft consonant). The only catch is that it is longer than other consonants, much like pronouncing two: [ɕɕ].
If you are accustomed to the other soft consonants in Russian (like ‘ть’, ‘сь’, ‘нь’), then ‘щ‘ shouldn’t be very hard for you. Simply palatalize your English ‘sh’ and make it longer!
How is ‘щ‘ different from ‘ш‘? ‘ш‘ is a hard, retroflex consonant. It is pronounced like ‘sh’ in English ‘ship’, but it is a retroflex consonant, meaning that the tongue is in the same shape as the American English ‘r’. It is transcribed as [ʂ] in IPA.
Here are some example words for you to study. Try to notice the difference between ‘ш‘ and ‘щ.’
Hard ‘ш’:
Soft ‘щ’:
therussianblog 
Wonderful! Finally someone who provides accurate information rather than “kind-of-sort-of-more-or-less-like, but…” garbage! The IPA should be taught at school, if you ask me.
Some Russians palatalize щ
in such a way that air is expelled along the path where the cheeks meet the teeth.
One way to achieve this is to draw the tongue back, flattening the back of the tongue, as if you were trying to make the sh sound on your soft palate.
bullshit
Thanks for the explanation! An indirectly related question: how does one pronouce the second person termination шь? Does soft mark ь affect the consonant, or is it pronounced exactly as if the soft mark were not there? I believe I’ve seen soft marks after other consonants which are always hard, and it puzzles me somewhat…
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