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Conceptually, I understand the difference between underlying (?phonological), surface (?phonetic) and lexical (?) "levels", but what are these levels?
I think they are just a heuristic, a sort of scientific shortcut on which all linguists agree, because they have explanatory value... but I'd like your expert input.
Based on your comments, I think your question is narrower than just "what is a level", it's primarily about derivational levels in phonology. Greg Lee's answer correctly identifies the essential property of "level" as applied throughout grammatical theory, and in phonology we have added on a few more concepts. Your question (as applied to phonology) also points to two distinct concepts of "level" in our technical vocabulary.
One sense of "level" is "substring", in autosegmental thinking. Any utterance at any derivational stage has a number of "levels", so that you can talk about the word, syllable, and featural levels of a representation. It is not a precise technical term, but people will say things like "This consonant is final at the foot level", meaning that the consonant is the last segment in the foot, even though it is not final in the word. It corresponds to "a node of type N" in representations, given that there is said to be a set of node types that make up representations (Place, Vocalic, Root, Syllable, Foot, Tone...). This sense of level isn't generally applicable to SPE style phonology, excluding the bits that are not clear regarding morphemic features.
The other sense is derivational, which is orthogonal to the question "what does a phonological representation look like". In classical generative phonology, a derivation is a chain of representations mapped to other representations, via the rules of a language. The initial representation is the underlying forms and the final representation is the surface form. We don't use the term "deep structure", which was only used in syntax and I think the concept was officially renounced as part of the Minimalist Program (and in other streams of syntax earlier than that). In classical generative phonology, all representations are phonological and there is no such thing as a phonetic representation. There can be any number of "levels" between the input and the output, each corresponding to a string mapping (i.e. what a rule does). None of these levels are significant.
Eventually, people opted to say that there are some significant levels. The (historically) first of these was the addition of a "phonetic" level, which is based on the realization that phonological rules cannot actually handle phonetic implementation. This more or less corresponds to the distinction between regular phonological rules versus allophonic rules, with the caveat that a number of "allophonic" rules probably are phonological rules which create new phonemes. (Flapping is an example). The theory of lexical phonology then created the potential for many new significant levels, also called "strata".
Sharon Inkelas in her dissertation provides a straightforward method of reducing derivational level and representational level to just representational level. It's actually completely obvious: take a phonological string, with stratal bracketing (the derivational concept of "level"), and let each stratum be a node in a new tree structure (α,β,γ) so that Level 1 is "anything under the same α", Level 2 is "anything under the same β" and so on.
OT (via Candidate Chains) has essentially reduced the full concept of "derivation" to "representation". In a classical derivation, you might have something like: /bunt/ → bund → bunid → [bunud], where the underlying form is /bunt/ and it is pronounced [bunud] by applying post-nasal voicing, epenthesis and vowel harmony. But this can be subsumed under the notion "surface form" where you have (bunt,bund,bunid,[bunud]) and the last item is what it pronounced (plus there are constraints on well-formed adjacent substrings within this "candidate").
There is a much narrower question about "levels" that could also be investigated, namely the utility of the Lexical Phonology concept of "stratum".
A level is a grammatical representation of a language expression. Sometimes stratum is used to mean the same thing.
Transformational grammar is multi-level, or multistratal, since in the transformational derivation of a language expression, each transformation that applies gives you a new level. Many transformations are involved, so there are many levels for each expression.
Although there are many levels, the levels are all the same types of thing -- trees. There was a controversy between transformationalists and stratificationalists (like Sidney Lamb) who argued that there were several levels, only a few, but they were of disparate types. This seems also to have been the view of the American structuralists, who complained that transformational theories were guilty of "mixing levels".
On the other hand, Generalized Phrase Structure Grammar, GPSG, is a single-level or monostratal theory. Any given language expression is described by a grammar with a single tree structure, not a sequence of trees as in TG.
It is conceptually possible to have a monostratal grammatical theory that includes phonology as well as syntax. To my knowledge, I am the only one to have proposed such a theory, here: Eliminating intermediary forms ....
