This is a very general question and one that can be answered on a number of levels.

First, it there is perhaps an explicit that second language acquisition is some specific situation, such as classroom learning as opposed to immersion learning. Then there is the related question as to whether we are talking about conscious learner strategies or teaching methodologies, rather than unconscious cognitive processes.

Related to this is the question of whether the language is being learned monolingually or through the medium of or with the assistance of first language, or some other second language. Similarly there is the question of whether the language is being learned solely orally or through the medium of or with the assistance of reading and/or writing. If reading/writing is involved, there is the further issue of whether the orthography is common with the FL, and whether it phonetic or idiographic or some kind of mixture of both (as many Asian languages).

This series of binary questions leads to hundreds perhaps thousands of possible learning scenarios before we even get to individual differences, including previous exposure to one or more foreign languages (particularly as a child).

It would be helpful for you to include example citations of the "language analytic abilities" being tested/measured (in particular what are these tests and what assumptions do they make in relation to the dichotomies I've outlined.

My approach would be to ignore all of the educational methodologies and learner strategies that are applied consciously as not being related to language analytic abilities or cognitive linguistic processes. But from the miniscule information in the post, I suspect an Educational Psychology focus.

First consider a child learning a mother tongue or multiple familiar languages (in the sense of pertaining to the family and extended family, their villages, and their political and trade environment). Outside of bastions of English monolingualism in the US, Australia and England, second languages are likely to be learned largely by osmosis just like the first languages. In any case, if a child is just exposed to language in school with non-native teachers, they are unlikely to actually learn it in the sense of obtaining a significant degree of competence, but even short trips to an appropriate country or region can have a huge impact.  The school training can help with vocabulary learning and give a head start making sense of the grammar and phonology, but in the end interaction with native speakers is required (in the absence of native speakers what is learned is some sort of pidgin or creole - which sometimes goes to have an influence back on the original language as these people write influential works). The availability of AV material produced by native speakers, and watching (say) movies in English without subtitles, can act as a surrogate for native speaker interaction to an extent.

So given I am discounting the teacher-mediated, first-language-mediated, classroom-bound language teaching as having much relevance to second language acquisition, I would be focussing on first language acquisition and immersive second language acquisition, including bootstrapping by watching videos and reading books in the target language.  I would further adopt a Cognitive Linguistics viewpoint and say the cognitive mechanisms were not solely or primarily linguistic in nature, but the same mechanisms that allows a child to learn about their environment (the physical world, the linguistic environment, the cultural environment, the social environment).  

There is strong evidence that young children do not distinguish between what is an arbitrary linguistic convention and what is a physical law, or something in between. There is a question as to whether this early plasticity, this ability to learn, is gradually lost (between 6 and 16) or whether the problem learning a second language has more to do with interference from the first language.  Those that learn a second language early enough seem to have a second 'box' for the second language. Even later learners may have such a 'box' but often all other languages are kind of mixed up in this 'box'.

So what are the cognitive mechanisms - basically recognition of similarity and contrast (the ideas that underlay Pike's theory of phonemics and his later generalization to tagmemics, the ideas that underlay Piaget's sticky mirror model, the ideas that underlie Cognitive Linguistics). There is of course also a strong role for memory of different sorts, including having the appropriate conceptual hooks available for linking the semantic and pragmatic concepts to. The Sapir-Whorff idea that culture and language influence and constrain each other explains why the wrong hooks being present, and the right hooks being absent, makes language impossible to learn in a classroom context focussed on language - real life learning is required to learn the cultural hooks.

Hi James,

If you refer to language analytic ability in the sense that  Skehan (1989) proposed, with reference to the Carroll's model of language aptitude (MLAT, 1959), then analytic ability involves both grammatical sensitivity and inductive language learning ability. Grammatical sensitivity,  as a top-down process, refers to the capacity to identify functions of words in sentences (e. g. subject, object, etc) whereas inductive language learning ability is a capacity to induce rules and create new sentences.  Other researchers working in the field of second language acquisition use the term in the same sense, for example Sternberg (2002), Robinson (2002), etc.  As far as I am aware, this has nothing to do with the conscious learning of  pedagogic rules in a second language although some kind of training and experience with learning more languages might have some influence on one's analytic ability.