I think the mean of the grammar of a language depends on whether you are a native speaker of this language or not.
In english grammar, I think for you dear NES friends, as riverkid said grammar is unconscious, I can say it's a reflex.
But for us, it has different meaning, it is one of the aspects of a language like listening, reading, etc. We need to study it to learn it, but you just can live normally to receive the sense of the grammar.
For non-NES, it also depends on the learner, for example for me, I do make grammer mistakes quite often but wouldn't say that I make them much more than the other ESL learners. Iíve never been taught grammar in detail; I didnít study by myself either. I donít know which one is past perfect, which one is something continuous tense etc(I don't know in my own language too.. For ex: I sometimes make passive sentences without thinking but if somebody ask me to convert one sentence from active to passive it takes long time to do it.Just my fingers type the letters unconsciously, I donít know whether they are right or wrong. But still this unconsciousness is different from your one, as I always think that "did I make a grammar mistake"?, I believe which's never been a thought of a NES. But on the other hand some Non-NES people know about the grammar even mora than the NES, they know the logic, rules and name of the rules.For them grammar must be a thought. But for you dear teachers, I think grammar has completely different meaning, which I'm not capable of to describe.