Assimilation is a phonological process in which a sound becomes more similar to a neighboring sound, copying features of place, manner, or voicing. Examples include nasal assimilation (input /n/ becomes [m] before /p/) and voicing assimilation. Assimilation may be progressive, regressive, or mutual, and can be context-restricted.
From your study of phonological features and phonological rules, you know that sounds are not atoms — they are bundles of articulatory features like [+voiced], [+nasal], [labial], and [coronal]. You also know that phonological rules operate on these features, changing sounds in systematic, context-sensitive ways. Assimilation is the most pervasive class of such rules: a sound becomes more similar to a neighboring sound by copying one or more of its features. The motivation is articulatory efficiency — transitioning between two adjacent sounds is easier when they share properties, reducing the muscular adjustment required at the boundary.
The clearest examples involve place of articulation. English has a prefix *in-* meaning "not" (as in *indirect*, *impossible*, *illegal*). Notice what happens: before /d/ we get *in-* (coronal nasal), before /p/ we get *im-* (labial nasal), and before /l/ we get *il-* (lateral). The underlying form is /ɪn/, but the nasal assimilates to the place of articulation of the following consonant. Before /p/ (a bilabial stop), the nasal becomes [m] (also bilabial). Before /l/ (lateral-alveolar), the nasal fully assimilates. The nasal "borrows" the [labial] or [coronal] feature of the upcoming sound. This is regressive assimilation (also called anticipatory): the following segment influences the preceding one.
Progressive assimilation works in the opposite direction: the preceding sound influences the following one. The English plural suffix provides a classic example. The underlying form is /z/, but after voiceless consonants it surfaces as [s]: *cat-s* [kæts], *dog-z* [dɒgz]. The voicing value of the final consonant in the root propagates forward onto the suffix. You can describe this with a phonological rule you already know how to write: /z/ → [s] / [−voice] ___ (the suffix becomes voiceless when it follows a voiceless segment). This is assimilation of the voicing feature.
Total assimilation — where a sound becomes completely identical to its neighbor — appears in phrases across many languages and in fast, casual speech. In Latin, *ad-* (toward) became *ac-* before /c/ (*accede*), *ap-* before /p/ (*append*), *af-* before /f/ (*affirm*). The prefix consonant fully copied every feature of the following consonant. You can see the same process in English loanwords and even in rapid speech: "in case" → [ɪŋkeɪs], where the coronal nasal assimilates fully to the velar place of the /k/. Assimilation is not a corruption or irregularity — it is a regular, rule-governed process that operates predictably across contexts, and your ability to write formal phonological rules from the previous topic is exactly the framework needed to describe and predict where it applies.