In a linguistic context IPA originally stands for International Phonetic Association, an organization set up in the 1880s for discussion of phonetic matters, and the standardization of phonetic transcriptions. Among the founders was Paul Passy, and another was Henry Sweet, the great phonetician that Shaw's Henry Higgins in Pygmalion is based on.

They devised a phonetic alphabet, called naturally enough the IPA alphabet. However, invariably the letters IPA are also treated as meaning this, the International Phonetic Alphabet.

In their early days they tried out quite a few different systems of transcription over a short period. Their journal was called The Phonetic Teacher when it was issued in 1886, but was titled and printed all in phonetic notation, so from month to month would come out with a name that was some variant such as Dhi Fonètik Tîcer. It became Le Maître Phonétique in 1889. Although the IPA alphabet settled down quite soon, they continued printing the journal in IPA until around 1971, when it became the Journal of the IPA.

There was a significant revision of the alphabet at the Kiel Conference of (?)1991. A number of new symbols were added, some terminology was revised, and the entire system of marking tone was changed.

One of the principles of the IPA is that you don't have to indicate super-fine shades of sound unless you're drawing attention to them. A language with five vowels roughly evenly spaced, such as Japanese, can be notated with the simple romanic symbols [a e i o u] even though there are special symbols that are more accurate. The square brackets around characters indicates that they are phonetic symbols.

Aesop's fable of The North Wind and the Sun has been used as a standard sample, so an old IPA handbook had it in about fifty languages rendered in IPA, from Icelandic to Korean to Xhosa.

The unofficial standard for mapping IPA into ASCII is called SAMPA, a simpler version for most European languages, and X-SAMPA, an extension to cover all others. It is primarily maintained by University College London (UCL).

By the way, the proper way to display IPA symbols as such on the Web, which however will not work fully on E2, because write-ups here don't support the <font> tag, is to enclose HTML numeric symbols in a font that many people are likely to have and that includes IPA. The recommended one for widest coverage is Lucida Sans Unicode, and another is Arial Unicode MS. So
<font family="Lucida Sans Unicode">['st&#633;&#594;&#331;g&#601;]</font>
should display a phonetic rendition of the word 'stronger'. If I use that in this write-up, however, what you see is:

Using IPA in Unicode: