'Schwa' is used both for the IPA symbol ə (HTML code ə) and more generally for the sound itself, as a synonym for neutral vowel. By neutral is meant that it is midway between the highest and lowest, and the furthest forward and back, positions that the tongue reaches in making vowels.

The term 'schwa' originates in Hebrew grammar, where it could also be spelt shewa or sheva or shwa. It refers to the sign of two dots one above the other, placed under a consonant to indicate either that it had no following vowel, or that it had a short following neutral vowel. The choice of which it is depends on syllable structure in a rather complicated way I'm not confident enough to explain.

It also occurred combined with one of several other vowels, such as seghol for a mid-open /e/. This combination is called hateph-seghol. This is traditionally described as an extra short seghol, but it might actually have been a vowel that could be either neutral or seghol, e.g. as you get fluctuation in English 'remain', 'November': some speakers, at least in some situations, use a short "i" or "e" vowel in 'remain', others or at other times use a true neutral vowel.

These fine details of vowel sounds were laid down when the Masoretic pointing was devised for Hebrew, a thousand years or so after the consonant texts had been written. Linguists disagree over how to relate them back to the living language of Biblical Hebrew.

In English the schwa is the first and third vowel in 'banana' (I think few people would use an actual "a" vowel there), or the U in 'circus', or the O in 'bigot'. With E and I it is more dependent on dialect: in most American and Australian speech the unstressed E of 'remain' or I of 'horrid' are also neutral vowels, but in some varieties of British, especially in Received Pronunciation, they are a short I, so that 'livid' has both vowels the same. In the non-rhotic accents the unstressed sequences containing R as in 'letter', 'beggar', 'victor' are also schwas.

In most languages where it occurs, the sound is confined to unstressed positions. In Bulgarian you can have stress on neutral vowels.

The schwa sign is used as a letter of the alphabet in Chechen; and also in the new romanization of Azeri, as in Azərbaycan. The capital form is a bigger upside-down e, not an upside-down E. However it is not pronounced as a neutral vowel, but as a front vowel [æ] and was formerly written ä.

(2004) I've started using the HTML code lately because browsers these days seem to show it even if they don't show most IPA symbols. The SAMPA (7-bit ASCII) equivalent of ə is @.