In phonetics, this means the coalescence of the Y-sound ('yod') with a preceding consonant to make a different sound. Specifically it refers to the phenomenon in English where alveolar consonants t d s z change into postalveolar ones ch j sh zh.
Historically this happened hundreds of years ago in some positions: the -tion and -sion endings have had the sh zh sounds for something like 500 years. They were originally pronounced -syon and -zyon when the words were borrowed into Middle English from Middle French. But these coalesced into -shon and -zhon long ago.
The next stage was in words like 'picture', 'nature' where the change from -tyoor to -choor was probably complete some time in the nineteenth century (and then became -cha); and likewise 'fortune'. The pronunciations fortyoon and piktyoor were apparently still heard around 1900, but Fowler castigated them as academic, and said the words were clearly forchoon and pikcher.
The discussion so far has been of sound changes common to most or all varieties of modern English. The term 'yod coalescence' is not a standard one in linguistics, but was coined recently: in 1982 by John Wells, the leading British phonetician. He used it to mean the twentieth-century sound change in some varieties of British English and related accents such as Australian, in which tyoo etc become choo etc.
Coalescence took place first in unstressed syllables; FORchoon, intelLEKchooal, GRAjooal. A few people still regard this as vulgar or incorrect, but it's pretty frequent.
The thing that really gets up a lot of people's noses is the more recent change in stressed syllables: 'tune' as choon, 'duke' as jook, and so on. This is one of the features of Estuary English, the accent that is encroaching on the older British standard, RP. Indeed, it's now common among announcers on the BBC's classical music station Radio 3, so choon is definitely standard English now.
This change shouldn't occur in those dialects in which tyoo had previously changed to too: in most American accents there is no longer a yod in 'tune, duke, new, suit', so nothing to coalesce with. Likewise in some accents of eastern England, the same pronunciations toon, dook, noo are used. This loss of Y has been dignified with the name yod dropping.
Except that it has occurred in Cockney: which formerly had toon, dook, but it is reported that very quickly the alternative pronunciations choon, jook are supplanting them.
With S the situation is a bit different. 'Sugar' and 'sure' have long ago changed their sy- to sh-, but 'sue' is, for most of us, now soo rather than syoo. This is not a UK/US dialect difference, but is more a variation in the individual words. Older speakers still say syoo but it shows no sign of changing to shoo in any more words: Perry White said Syooperman and most of us now say sooper. For 'assume' some say asyoom and others ashoom, and Americans and East Anglians asoom.
The change may also take place across close word boundaries: 'would you' = woojoo, 'can't you' = kaanchoo.