Oh, the primordial soup: the deep stuff.
First off, I'm not sure "awareness", the state of being aware, is much different from "consciousness", the state of being conscious; the way the words are typically used, awareness is more refined consciousness (assuming a phrase like "she was barely conscious, not really aware of anything" is fairly descriptive), so that we can attend to some content of consciousness long enough to process it: compare, identify, associate, expand, repair, enjoy, etc.
Maybe start with cases. When learning a skill like driving, initially my awareness of the skill is as a sequence of steps. Step 1: start the car -- is learned as a sequence of explicit, either verbal or diagrammed, steps: locate ignition key, determine 'right-edge up', slowly insert key 'right-edge-up' into ignition, turn key clockwise until starter engages, allow key to return to insert position, repeat if engine stalls, etc. We are aware of each step separately at first, then less and less so as the skill is acquired and the steps are integrated into one larger step, starting the car, within the skill of driving.
It seems then that awareness has different intensities: the awareness of one step diminishing until it effectively disappears within a larger step. Which relates to "understanding", the second undefined term. From the example, I think it's fair to say that awareness decreases as understanding increases, until one understands the process so well that she performs all the steps within it without being aware of them individually.
Is this true of other activities to which the word 'understanding' applies? When one learns to do multiplication, for example, does one's awareness of the steps decrease as one's understanding of the process increases? That's a huge question, and could easily fill a book, I think (which I don't have the expertise to write). Off the top of my head, it seems here "understanding" has to do with making the process mechanical. At first, the student understands multiplication as: (xxx)+(xxx)+(xxx)+(xxx) -- which she is told to write "3 x 4" = 12. With memorization of the multiplication tables, that visualization step disappears: 8 x 9 = 72; there's no sense diagramming it as (xxxxxxxx)+(xxxxxxxx)+... rpt 6 ...+(xxxxxxxx), to make the student aware of the complete 'logical' translation for that step, as that step is already understood within the mechanics of the multiplication table. (Ideally, that is. It occurs to me here that for many students multiplication, or rather math in general, may be understood strictly as mechanics to be learned by rote and regurgitated by ritual, rather than as descriptions of ideally ordered and described systems. In which case, lack of awareness, of the logical level, equates to lack of understanding. I think where the student is made aware of a diagram for the process before equivalent symbols, "understanding" may be said to correspond to the student's ability to reproduce those original diagrams if necessary, not to be constantly aware of them while manipulating the symbols. But as I said, we're on the border of book-filling territory).
So before this starts to look like a book, I'll tentatively offer these definitions: "awareness" is the place of mental data prior to integration into largely unconscious process(es); "understanding" is a measure or how successfully one has integrated data for use by these larger unconscious processes (algorithms, on the computational model). A last brief example: it's interesting that when one has learned a skill, like a golf swing, and then wants to improve that skill, that one aims to divulge and dissect the original learning steps within the skill; i.e., make them conscious again, so to reintegrate them more successfully, to "break bad habits", as the saying goes. Part of the role of consciousness seems to be as a sort of clearing-house, where bits that haven't been integrated yet, or need to be re-integrated, are held until some unconscious urge spots a need for one or some of them, ponies up the neurochemical dough to adopt it into memory, integrate into process, make it one of the family, give it a go, <insert your own apt metaphor>, etc.
There. Football's on... (my saints now aware that the falcons are in fact they who dat believe they understand how to beat them, or words to that effect).