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A nasal vowel is a vowel that is produced with a lowering of the soft palate (or velum) so that the air flow escapes through the nose and the mouth simultaneously, as in the French vowel /ɑ̃/ (help·info) or Amoy [ɛ̃]. By contrast, oral vowels are produced without nasalization.
Nasalized vowels are vowels under the influence of neighbouring sounds. For instance, the [æ] of the word hand is affected by the following nasal consonant. In most languages, vowels adjacent to nasal consonants are produced partially or fully with a lowered velum in a natural process of assimilation and are therefore technically nasal, but few speakers would notice. That is the case in English: vowels preceding nasal consonants are nasalized, but there is no phonemic distinction between nasal and oral vowels, and all vowels are considered phonemically oral.
The nasality of nasal vowels, however, is a distinctive feature of certain languages. In other words, a language may contrast oral vowels and nasalized vowels phonemically. Linguists make use of minimal pairs to decide whether or not the nasality is of linguistic importance. In French, for instance, nasal vowels are distinct from oral vowels, and words can differ by the vowel quality. The words beau /bo/ "beautiful" and bon /bɔ̃/ "good" are a minimal pair that contrasts primarily the vowel nasalization even if the /ɔ̃/ from bon is slightly more open.
Although there are French loanwords into English with nasal vowels like croissant [ˈkɹwɑːsɒ̃], there is no expectation that an English-speaker would nasalize the vowels to the same extent as French-speakers or Portuguese-speakers. Likewise, pronunciation keys in English dictionaries do not always indicate nasalization of French or Portuguese loanwords.