Assimilation is the most common form of sound change, and so it's a good idea to spend time on it. The essential idea is that sounds in proximity to one another can change one another, and they can do so when they are right next to one another or when they are scattered throughout a word (or even a sentence - pepper becoming 'pecker' in 'Peter Piper picked a peck of pickled peppers' is an example of assimilation at a distance). It's extremely common and found in all languages and language families. It's different from fusion, in that the sounds do not fuse together to become a single sound. Instead, they change one another slightly (or completely) but maintain their separateness.
So how does it work? It's not so tricky if we bear in mind the concept of feature (again). Like fusion, assimilation depends on feature. Here, though, we need to introduce the concept of phonetic similarity. This is when two sounds share features, as you can probably infer from the name. As Crowley notes,
If a sound change results in an increase in the number of shared features, then we can say that assimilation has taken place. (p.37)This is fairly simple: [m] and [n] are similar sounds in that they are both nasals (sonorants). [p] and [d] are similar in that they are both stops. Since sounds have lots of features, and not just one, there are many ways in which they can assimilate. When sounds assimilate only some of their features, and not all, it is called partial assimilation. If they assimilate all the sounds, it is total assimilation, or gemination, in which the sound is doubled (see below). Crowley's example for the general concept of assimilation is [n] and [p]. Put [n] and [p] next to one another, np, and you can expect certain sound changes to happen to [n] (or to [p] - see below). (You don't need to know exactly what these features are, but it would certainly help, and a visit to wiki's page on distinctive features might be a good idea.)
If the sonorant (nasal) feature of [n] disappears in assimilation when next to [p], then it will become a stop - [d]:
*np > dp
If the place of articulation of [n] assimilates, then it will become [m] ([m] is labial, i.e., its place of articulation is the lips, the same as the place of articulation of [p]):
*np > mp
Crowley also lists an example that isn't too common, but which is found in some languages, especially in south Asia. That's the assimilation of the voiced feature of [n] to become voiceless. This is represented by [n̥] - a voiceless alveolar nasal. So the sound change is:
*np > n̥p
As it is possible for multiple features to assimilate, this is not a complete list, and [n] could become [b] (retaining the voiced feature of the [n] and nothing else), [t] (keeping the place of articulation of [n] and nothing else), or [m̥] (retaining only the nasal feature and assimilating the place of articulation and the voicelessness of [p]).
If all features are assimilated, then we have a case of gemination or total assimilation as mentioned above. A good example of this is in Pali, an Indo-European language formerly spoken in what is now India and Nepal, and which is the liturgical language of Theravada Buddhism. Pali is closely related to Sanskrit and is found alongside it in many contexts. This means that assimilation is fairly obvious (we have a closely related language to compare it to which did not undergo the same sound change) and is a regular rule in Pali. This is also useful in understanding the evolution of other Indo-Aryan languages, many of which exhibit similar sound changes (and further ones, as well).
'action' or 'deed'
Sanskrit: nirva:ṇa (note the voiceless alveolar nasal)
'profound peace of mind'
This is all regressive assimilation - i.e, the change occurs in the phone preceding the one that influences it. A <---- B Progressive assimilation, which is rarer, occurs in the opposite direction. A ----> B
Crowley then lists all the sound changes that can occur if progressive assimilation applies to the same [np] cluster - i.e., *np > nn (total assimilation), *np > nb, *np > nd, etc. There is no need for me to go through the list, and if you want it, then buy a copy of Crowley and Bowern, which would be a sensible decision on your part anyway.
Crowley's example for partial regressive assimilation (the most common kind) is, characteristically, from Australia. In fact, it is a Pama-Nyungan example, looking at the Karnic languages.
Here, in Yawarrawarrka, the stop feature of the [p] has been copied such that the lateral [rl] has become a stop, [t]. Yandruwandha and Diyari are unchanged.
Crowley's example of total progressive assimilation, one of the least common kinds, is less characteristically from Icelandic (the fact that he's using one of the canonical languages of the Indo-European tradition shows how hard it is to find examples of this!):
*findan > finna 'find'
*gulθ > gull 'gold'
*halθ > hall 'inclined'
And so on.
One important type of assimilation is palatalisation. This is important, and quite well-known, both because it is fairly common and because it explains an important split in the Indo-European languages: the division into centum and satəm languages, derived from the words for 'hundred' in Latin and Avestan respectively. The [k] became an [s] through palatalisation (with an interim stage of [ʃ], a voiceless domed post-alveolar sibilant) in some Indo-European languages and remained a [k] (or similar) in others, the split occurring deep in prehistory.
Palatalisation is an assimilatory change by which non-palatal sounds become palatal sounds. This is common before front vowels, like [i] and [e], and there are lots of examples, both sporadic and regular. One regular change is the palatalisation of [d] to [d ] in Portuguese de, and similar changes in Fijian before [i] - the name of Fiji in Fijian is in fact Viti, pronounced 'Fiji'. 'Petula' contains an example of palatalisation as well (at least, the way I pronounce it!). The word hue is also a good example, especially as many speakers of English are unaware that they have palatalised the [h] to become [ç] (a voiceless palatal fricative). Due = Jew is another example. The history of English is replete with examples, actually; *kinn > chin, *geldan > yield, *gearn > yarn, etc.
I will post some more on assimilation in a day or so. There is of course a little more to it, and there's also the concept of dissimilation to cover as well. After that, we've got tone acquisition to get through, some unusual sound changes that don't fall into these categories, and then we're done with types of sound change. After that, there's a test on how well you understand the principles, which I will go through in minute detail - if you intend to buy the book then I suggest avoiding that post. Then, we're onto the representation of sound changes (this'll help unlock the linguistic literature for you) and then there are all the practical matters of acquiring data, assembling tables to check for sound correspondences, and other things that make historical linguistics an interesting and important method for the human sciences. So stay tuned.