The centum-satem isogloss divides the Indo-European languages from a pretty ancient stage in their development. Essentially, the split concerns the treatment of the Proto-Indo-European velar consonants. The traditional analysis is that PIE had nine of them:
In the centum languages, the palatovelar series merged into the plain velar series. This group includes the Celtic, Italic, Germanic, Hellenic and Tocharian (although Tocharian did it a little differently) branches of the Indo-European family.
In the satem languages, the palatovelar series became sibilants, and it was the labiovelars and plain velars that merged. This group includes the Balto-Slavic, Albanian, Armenian and Indo-Iranian branches.
The PIE word for “hundred” began with one of the palatovelar consonants (*ḱ) and so the name of the split comes from two early reflexes of that word showing how the words changed in descendant languages. Latin centum /kentum/ demonstrates /k/, and Avestan satem demonstrates /s/.
Originally, the centum-satem isogloss was posited as a west-east split, but the later discovery of Tocharian forced a rethink on that, being the easternmost Indo-European language discovered and yet also being, apparently, a centum language. It’s now thought that centum-satem is more of a periphery-centre split, with the branches that remained spoken for longer close to the PIE “homeland” passing the satem innovations between each other.
There is some debate in modern times about whether the traditional description of PIE velars is the most likely possibility. An alternative theory is the “uvular theory”, whereby the “palatovelars” were actually plain velars, and the “plain velars” were pronounced further back, possibly as uvular *[q], *[ɢ], *[ɢʰ]. So basically, the velar consonant chart would instead look like this:
In this theory, at some point the uvular series fronted to plain velars *[k], *[g], *[gʰ]. In the satem branches, this caused a chain-shift where the existing plain velars palatalised. In the centum branches, no shift occurred and the uvulars and plain velars merged, instead. In this theory, the delabialisation of the labiovelars in the satem branches occurred later, separately. Some of the reasons that this alternative theory seems likely are:
- The traditional “palatovelar” series are extremely common phonemes in PIE, while the traditional “plain velars” were pretty uncommon and not used in any affixes. Cross-linguistically, it would be a lot more typical for the plain velar series to be most common.
- There’s no residual evidence in the centum branches of any of the “palatovelar” reflexes ever having been palatal, and because the centum branches aren’t thought to have shared a common ancestor that they didn’t also share with the satem branches, the traditional theory would require a pretty improbable simultaneous depalatalisation in every centum branch, and none of the satem ones.
- PIE is reconstructed as having an extremely rare phoneme /a/, which nearly always occurred adjacent to /h₂/ or /k/. /h₂/ is basically incidental to this discussion, but if /k/ was actually [q], then the appearance of this /a/ in the reconstruction could be explained as [q] having an “a-colouring” effect (basically back vowels like [a] are way easier to pronounce adjacent to [q] than other vowels). In modern languages that have /q/ as a phoneme like Quechua and Arabic, it has indeed been observed that vowels tend to move further back next to /q/.
Many Indo-European languages have seen further sound changes on reflexes of PIE velars. For example, the very /k/ in Latin centum has become /s/, /tʃ/, /ts/ or /θ/ in virtually all of the modern Romance languages, but languages’ classification depends only on what happened to the velars way back then at the time of the split and not what happened to them afterwards.