Let us first consider a creature that is sentient but not self-aware; it is conscious, but does not know it. The experiences of this creature are restricted to sensations from its sensory organs, such as touch and smell, and any internally generated feelings or emotions such as fear and joy. This creature cannot ponder these sensations; indeed, what it lacks is the ability to separate itself from the feeling and think "I feel that".
Humans, and it seems likely to me that in this we are not alone, do have this sense of 'I' and we do distinguish between the sensations we experience. Using the example of happiness: rather than simply feeling happy, you know that you feel happy. One can say "I know that I am aware" however, interestingly, one can also say "I know that I know that I am aware" and indeed "I know that I know that I know that I am aware". This reflexivity could go on forever, yet would not add any more meaning to the base statement "I know that I am aware".
What we see is that to be self-aware is to be aware of one's own thoughts and experiences. One has an experience and is aware of it, this enables one to think specifically about this experience. Furthermore, one is also aware of one's own thought about this experience, which is where the reflexivity comes in. I therefore define 'self-consciousness' as 'to be aware of one's own conscious experience', thus including reflexivity (one's thoughts about one's thoughts).