In Indo-Iranian and Balto-Slavic languages (which are conveniently all Satem languages), there is a sibilant or affricate sound in places where Centum languages usually have a velar consonant. It seems reasonable for Centum languages to simply depalatalize, voice, lenite, or even glottalize the /ḱ/, remaining mostly in the rear/velar area, as opposed to Satem languages shifting all the way to /ʃ/ and /tʃ/. For example Quattuor in Latin is Chetiri in Slavic and Chatur in Sanskrit. How did this come about? Is there name for a rule of this sound change?
The other answer focuses on the observation that it happens, and has a name; I address the "how" question. The starting point is having a difference between velar [k,g] and palatal [c,ɟ], the latter being a kind of fronted k,g. You can hear expert IPA demonstrations here. Although the articulatory difference between [k] and [c] is small, the acoustic effect is quite large. The palatals sound very much like linguals such as [ʃ, tʃ, ɕ]. This makes it easy to confuse true palatal sounds with postalveolars or alveolo-palatals, indeed the term "palatal" is used in many ways. As applied to IE reconstruction, the premise is that PIE started with [c, ɟ] (in IPA: /ḱ, ǵ/ in more traditional Indo-Europeanist orthography), then the satem languages simply followed a natural phonetic tendency of alveolarization to arrive at [ʃ] etc. Incidentally, North Saami historically also has a distinct palatal series of consonants spelled dj, which are similar to but still phonetically different from Hungarian dy, and in some dialects of North Saami, they have by this same alveolarizatin process merged with the alveopalatals [tʃ] spelled č.
The same [k] → [c] → [tʃ] path is attested in numerous languages that did not start with [c, ɟ] as PIE is assumed to have. As for terminology, the broadest term is "palatalization" which typically refers to any back of the mouth to front of the tongue shift. There are very many types of palatalization, and phonological studies there is a tendency to distinguish the alveolarization step, which is motivated by acoustics rather than articulation. The step where [ki] → [ci] is articulatory, involving an adjustment to the velar target into a fronted velar a.k.a. true palatal. Once there, the sound is susceptible to further perceptually-motivated reanalyses, based on the acoustic similarity of [c] and [tʃ].
The example of "four" actually does not relate to the Indo-European palatals, since the PIE root has [kʷ], not [ḱ]. A better example of a PIE palatal is the word "hundred", with [ç] in Sanskit, [s] in Slavic etc – the Sanskrit outcome is earlier in the historical chain than the Slavic outcome is. This just underscores the point that this shift is very common across languages, and can happen many times in the history of a language.
As ngn says in a comment, this is palatalization.
It is a very common sound change. It happened in the descendents of the poster-boy centum language Latin:
Latin ''centum'' (/k/) -> Italian ''cento'' (/tʃ/), French ''cent'' (/s/), Spanish ''cien'' (/s/ in Latin America, /θ/ in Spain)
In Germanic it happened differently in different times and places: Danish ''kirke'' and Scots ''kirk'' have neither /k/ palatalized, German ''Kirche'' has only the second palatalized, Swedish ''kyrke'' has only the first palatalized, and English ''church'' has both.
Further afield, Arabic has no /g/ because where its relatives have that sound Arabic has /d͡ʒ/, for example Arabic جَمَل (jamal) vs Hebrew גמל (gamál) "camel".
And Beijing used to be written Peking because when Europeans first encountered the city the Chinese name had a /k/ in it, but over the last four hundred years the sound has been palatalized to /tɕ/.