The twenty-second letter of the
Greek alphabet, capital form Χ and lower-case χ , for which the HTML codes are Χ and χ . It was not part of the original
Semitic alphabet the Greeks borrowed.
In English it's pronounced [kaI], KY as in Kylie. In Ancient Greek it was [khi:] or [khe:] (the alternative spelling chei) and in Modern Greek it's [çi]: see below for details of pronunciation.
It had two values. In the western dialects it was used to represent KS, and this value passed to Etruscan and thus to the Romans, and it is so used in most European languages using the Roman alphabet. It is unclear why the Greeks wanted a single letter for KS, and also the letter psi for PS, but one pertinent fact about these two is that they were the only consonant clusters that could occur at the end of a word: onyx (later written ονυξ). To be more accurate, NKS could also occur, as in lynx.
The western value died out in Greek, and the classical language (Attic) and Modern Greek use the eastern value.
In the eastern dialects, another letter, xi, was used instead for the value of KS, and chi was used for the aspirated K sound. This had earlier been written KH, but the letter η (eta) ceased to be used as a consonant H and was used as a vowel. So the new letter chi was assigned to the value Kh, which is the sound of English K (or K-like C) when it begins a word: cat, kit, cut. It's followed by a distinct aspiration, or H-like quality in the early part of the vowel. In technical terms, it's an aspirated velar stop.
The English words scat, skit, scut lack this aspiration. In English it's conditioned by the presence or absence of the S, but in Ancient Greek they were quite distinct phonemes. Unaspirated K was written kappa and aspirated Kh was written chi.
In the Roman alphabet K was not used, so Greek words with kappa and chi were written with C and CH respectively. This sound CH did not naturally occur in Latin, but educated Romans knew Greek and used it. (Actually some native Latin words did develop the sound: e.g. pulcher 'pretty'.)
After the classical period, some time in the first centuries CE, this sound changed to a fricative value. In Modern Greek it has two different fricative values, depending on position. I am unsure whether these allophones developed later or at the same time as the shift from Kh to fricative. Today its two values are distributed similarly to the ich-laut and ach-laut of German, i.e. the thin SH-like sound beside I and E, and the guttural sound of Achtung, loch, chutzpah beside A, O, or U. However, it's the following vowel that decides it: Modern Greek okhí is pronounced with the ich-laut. (In German it's determined by the preceding vowel.)
I have a slight problem that there is no obvious neutral symbol to describe the guttural ach-laut, because of the history of the letter. KH, CH, and X are all ambiguous. The IPA symbol for the original aspirated value is [kh], that of the palatal ich-laut is [ç], and that of the velar ach-laut is a Roman [x]. The actual Greek lower-case letter is also used as an IPA symbol, [χ] being a voiceless uvular fricative. (Some texts, particularly older philological ones, use it instead of [x].) So Modern Greek okhí is pronounced [o"çi].
In later Latin CH would have been pronounced as C. In theory this should then have shifted to the various other sounds an ordinary C became in Latin's descendants French, Italian, Spanish, and so on, because it was just a C: cf. cinq, cinque, cinco. But what we in fact see is that CH is still generally a K sound, as in archetype. Probably this is because these are all learned words, so scholars could correct them to Greek values. In the Renaissance, as Greek started being taught again, it was taught by native Greek-speakers who naturally had the modern, not classical values. It was the work of people like Erasmus who initiated a rediscovery and restoration of classical values.
One notable exception is surgeon, from an earlier cirurgeon, from Greek roots cheir-ourg- 'hand-work-'. Clearly CH did become C and follow the standard sound-change to S.
Because English-speakers find it hard to make the K and KH sounds distinct outside their conditioned environment (skit vs kit), the advice to students has usually been to pronounce chi as [x], and similarly phi and theta with their modern English values. I personally disagree with this advice.
In modern romanizations of Ancient Greek, chi is sometimes rendered by KH, and in romanizations of Modern Greek it is usually rendered by KH or sometimes H, since the ach-laut sound can be rather weak: thus Khalkidhikí or Halkidiki for classical Chalcidice. This has the advantage that classical names have undergone many sound changes in their passage through Latin, French, and English, so English words like chalk and chalcedony are misleading.
In a few words people seem uncertain what to do with the CH, and give it a French SH-sound. I have heard this for both Chiron and chimera. I would strongly recommend a K sound in both. However, in the word chiropodist the quasi-French pronunciation seems to be well established.
The X-shape of the letter chi is the origin of Xmas for Christmas, and the less common Xtian or Xian for Christian. In the ancient Gothic alphabet of the 4th century CE, based on the current values of Greek, X is not used as a letter, except in the name of Christ. Although Gothic did have a chi-sound at the end of a syllable, written H, it is perhaps more likely that this initial X was pronounced K.