Pronunciation of "err"
There are two pronunciations:
/ɜr, ɛr/
Based on Gilbert and Sullivan lyrics in Ruddigore, in London in the late 19th century it was pronounced to rhyme with "her":
If I had been so lucky as to have a steady brother
Who could talk to me as we are talking now to one another —
Who could give me good advice when he discovered I was erring
(Which is just the very favour which on you I am conferring),
Webster's 1892 Dictionary agrees: err has the vowel of her.
However, nowadays, in the U.S. Northeast at least, I usually hear it pronounced "heir". Both pronunciations should be considered correct.
As others have mentioned, err has multiple accepted pronunciations: one with the same vowel as "her," one with the same vowel as "air". This variation seems to be confined to certain accents of North American English, though: in some other accents, only the pronunciation with the vowel of "her" is possible.
The pronunciation with the vowel of "her" is earlier. In Middle English, this was pronounced more or less as /ɛr/ (this was different from the vowel sound used in "air"). The "short e" sound shifted in quality to /ɜ/ (the same vowel sound as "fur" and "stir") in almost all accents due to a historical sound change that affected short vowels before /r/ followed by another consonant or by the end of a word (Wikipedia describes this under the name of the "Nurse merger"; it references John Wells's Accents of English). For example, consider the pronunciation of the word "verse": for nearly all English speakers, it rhymes with "nurse." This pronunciation with the vowel of "her" is used worldwide.
The pronunciation with the vowel of "air" seems to be more recent. As far as I can tell, it only became common after the merger in some North American accents of /ɛər/ with /ɛr/ (the Mary-merry merger). In non-merged accents, it is not possible for a word to end in /ɛr/. But in merged accents, it is, so apparently some people with the merger began pronouncing "err" as /ɛr~ɛər/ by analogy with the pronunciation of "error" /ˈɛrɚ~ˈɛərɚ/.
In non-rhotic accents (including most varieties of British English and Australian English) the pronunciation /ɛr/ is not phonologically possible, and I haven't heard of anyone using /ɛə/ (the vowel of "air").