Vtyjkyuk

It says that morphemes include un-, -ed, -ness, and re-, in addition to words like town or chair or computer, words that can't be broken down.

But computer comes from compute and -er. And compute comes from com- ("with, together") and putare, "to reckon," originally "to prune".

Then the word chair comes from the word chaere, old french chaiere "chair, seat, throne". From Latin cathedra "seat".

So in a sense, these words can be broken down further. Even the morpheme -ness can be broken down or derived:

word-forming element denoting action, quality, or state, attached to an adjective or past participle to form an abstract noun, from Old English -nes(s), from Proto-Germanic *in-assu- (cognates: Old Saxon -nissi, Middle Dutch -nisse, Dutch -nis, Old High German -nissa, German -nis, Gothic -inassus), from *-in-, noun stem, + *-assu-, abstract noun suffix, probably from the same root as Latin -tudo (see -tude).

Trying to get a better sense of what a morpheme is. Wondering if one could point to a good resource. I can break down the words further using etymologies, so not quite sure what they mean.

The most important fact about "morpheme" is that it is a claim about the state of a language as it exists at a specific time; it is a concept of synchronic analysis, not diachronic analysis (etymology).

There have been numerous attempts to "define the morpheme", i.e. give a succinct statement allowing you to know that this is a morpheme and that is not. One problem is that definitions of "morpheme" rely on the notion of "word", but "word" itself is a controversial concept. A typical definition of "morpheme" is that it is the minimal meaning-bearing unit of language, which presupposes that a morpheme can be assigned a particular meaning. If you take that definition to be axiomatic, then it means that invert, convert, pervert, subvert are all monomorphemic, because vert can't be assigned a meaning, even if it has an etymology. Some people are okay with imposing strong semantic requirements on word-composition, others are not. Our analytic methodology is not so refined that it's clearly right or wrong to call subvert monomorphemic.

The "minimal meaningful unit" definition does cover the majority of cases where people agree "that is a morpheme" – provided that you have a generous enough definition of "meaning". Iranian languages commonly have a morpheme -i~-e called Ezafe / Izafe (and other things), which serves as a syntactic linker between words within the noun phrase: it's a bit of a stretch to say that it has a meaning, but it certainly has a function. Indeed, you could define morphemes in terms of function rather than meaning, as long as you can appropriately define "function".

There does seem to be agreement that we don't want to define "morpheme" in such a way that "pl" is a morpheme in English, attested in "place, play, please, plow...". The intuition is that even though "pl" is a linguistic unit of these words (it's a syllable onset), it is not a "grammatical" unit. This raises the question of the scope of "grammar". As a phonologist engaged in research into the theory of grammar, specifically phonological computations, I will simply point out that defining "grammar" as "anything that isn't about phonology of phonetics" is just terminological hijacking. Even then, defining "morpheme" as "the minimal unit of grammar" simply re-packages the problem of "subvert" – does the grammar combine two elements to give "subvert", or is it a grammatical primitive? All we did in shifting the definition from being meaning-based to being "grammar"-based is remove one of the accessible diagnostics for dividing a string into substrings.

The concept "morpheme" has little technical utility in modern linguistics, even though it is a well-known term and we teach it in intro linguistics all the time. If you are interested in that intro-linguistics concept, it's a minimal meaningful unit of (synchronic) analysis, and I've pointed to some of the problems with the intro concept. I don't expect that clarity on the concept will be achieved within my life.

You are right that historically, those words are made up from separate units. Morpheme could be used in a historical sense; but it is usually used synchronically.
In present day English, compute does not break down into separate units - we cannot use pute on its own, and related words such as impute and repute do not share any discernable element of meaning. So compute is a single morpheme in present-day English.

Whether you regard computer as one or two morphemes is to some degree a matter of choice. Historically, it was formed from compute + -er, but it is not clear to me whether in the psyche of present-day English speakers it is analysed in that way.