Gosh, sorry I haven’t been checking this forum for the past week or so; I would have happily jumped in when this was a fresh thread. Looks like most of the questions have been answered, though.
One of the reasons there is confusion is that the terminology has changed over the past century or so. “Mute” is just a very old term for “stop” (or “plosive”), for instance. I also don’t think “sonorant” is used anymore: “resonant” is used instead, I believe.
I would provide a slightly different list from Raya’s:
Obstruent:
Raya’s description is pefect, except that I notice that nobody went on to define “affricate” (one of the sub-types of obstruent, which Raya lists). An affricate is something that starts out as a stop which is then immediately followed by a fricative sound. The English sound “ch” as in “church” is an affricate: it starts out as the stop “t” but then is followed by the fricative (and sibilant) “sh” (as in “show”). The English “j” sound starts out as a “d” and then it’s followed by “zh” (as in “vision”).
Approximant:
Raya listed glides (=semivowels) and liquids. I am not sure that “l” qualifies as an approximant since the tongue does make contact. Nor am I sure that all forms of “r” qualify. The kind of “r” that is used in English does, for sure, but something like a trilled “r” or even a single tap, probably doesn’t.
Resonant:
Quite simply, nasals, liquids, and semivowels (=glides). An interesting tidbit about all resonants is that in many languages they can become “syllabic” – i.e. they can be treated as vowels. Obviously the glides y and w can become the vowels i and u, but all the other resonants are often treated as vowels. For instance, in English, the second syllable of “rhythm” is (in many people’s speech) simply the resonant “m”. Most people would like to think that it is preceded by a short schwa, but that is not always true. Some people pronounce “mountain” with the second syllable being just the resonant “n” (usually, in that case, the “t” shows up just as a glottal stop: it’s hard to pronounce otherwise). And most Americans’ pronunciation of “little” will simply use a syllabic form of the resonant “l” as the second syllable.
Syllabic resonants are a very big part of the standard reconstruction of Proto Indo European and ablaut theory. None of the daughter languages have preserved the original syllabic resonants: they are either analyzed into some epenthetic vowel + a consonantal resonant (e.g. Latin “centum” where the “en” comes from a syllabic “n”), but a very common reflex of the nasal resonants especially is simply the vowel “a” (the hypothesis is that originally it was a nasalized “a” but that later on, the nasality was dropped): hence the Greek “alpha privative” (as in “athematic”, “amoral”), which began as simply “n-”, and explains why the Greek accusative singular in third declension nouns is sometimes “-n” (when it comes after a vowel), and sometimes “-a” (when it comes after a consonant): they were originally the same thing.
Syllabic resonants in Indo European studies are represented with a hollow circle underneath.
To get back to why so many of these terms are confusing: it is because they are not precise terms. If you want something more precise, you should learn about “distinctive feature theory”, which classifies sounds according to whether they do or not exhibit a certain feature, such as + or - (plus or minus) voice, ±rounding, ±continuant, etc. Still very valuable is the landmark book The Sound Pattern of English, by Noam Chomsky and Morris Hale, 1968. Distinctive feature theory was very successful in explaining many phenomena, including the ways in which children acquire language (it blew away Behaviorist theories of language acquisition, which make predictions that all turn out wrong). It turns out that sounds that share certain features will behave the same way in certain respects (sort of the way that elements in the same column in the periodic table will behave in the same way in certain respects).
The Sound Patterns of English is also a great book for understanding English Orthography, which turns out not to be as crazy as it looks. In fact, there was a study over 20 years ago that proved that non-native speakers of English, if taught English orthography through the theories from that book, can pronounce new English words they have never seen – often more accurately than a native speaker who was (mis)taught by our school system!