Just a note that the answer is going to vary depend upon the variety of English spoken. The page cited in teechar's post suggests 39 sounds, which seems a bit low - I've always heard it expressed as around 41-44 sounds.
The number of consonants is pretty stable between variants at 24, but it's the vowels which range from 15-21 sounds by variant. The exact number of voiced/voiceless consonants may fluctuate a bit by variant as well.
This
Wikipedia entry breaks down the numbers in a bit more detail by variant.
As an example, even in just AmE, there's variety in the number of vowels. Some speakers of AmE may hear the words 'Don' and 'Dawn' as the same vowel sound, while others hear them as two distinct sounds. The same goes for 'cot' and 'caught'. Speakers of the Western US English variant don't differentiate between the two sounds.
I personally can't distinguish any difference between the two pairs, so my English is going to have at least one less vowel sound than those who hear them as two separate sounds.
That's just one example from one variant of English. I'd imagine there are similar regional varieties for any given variant of English.