Often confused with the more specific term consciousness
People are supposed to be self-aware. That much everyone seems to agree on. As to exactly when that happens, there is much disagreement. Perhaps we were self-aware at the age of 6 months, but it doesn't seem likely - 6 month olds don't have much to indicate they are aware of themselves, or at least, themselves as different from anything external to them. Then again, maybe they just can't communicate it. Then there's the fact that almost no one really remembers being 6 months old. On the other hand, that doesn't mean you weren't self-aware... perhaps you just don't remember it.
As for what else may be self-aware, there is great disagreement, but most people believe that nothing is. Of course, most people are pretty stupid. Among more discriminating experts, whales, dolphins, monkeys and gray parrots frequently rate at least "some" self-awareness.
Self-awareness temporarily pauses during sleep, a nightly occurrence for most people. However, strangely, this often doesn't encourage much introspection about the nature of the whole thing. Sex, and in particular, orgasm, are considered by many to curtail self-awareness. The french refer to the latter (im)politely as the "little death."
Ah yes, let's not forget drugs. Drugs are typically agreed to be naturally occuring substances which happen to affect the functioning of the brain, either directly or indirectly. There are a million of them - humanity has had a great preoccupiation with them since the first one (widely regarded to be alchohol, though that's debatable) was discovered. Almost all "reduce" or "curtail" the functioning of the brain. Ironically, many drugs (not to mention sleep deprivation) are held to increase awareness and/or self-awareness.
This last effect is interestingly coincident with some religious beliefs. Many Asian and aboriginal religions, most notably Buddhism, hold that (roughly) when thought is eliminated (i.e. emptiness), a state of supreme enlightenment and transcendence is achieved. What this implies about the brain and self-awareness is equally suggestive and confounding. This, by the way, is the normal state of affairs when studying self-awareness.
Self-awareness is not widely believed to end in death - rather, most people seem to think there is some kind of life after death, where a person continues to be alive after all, except punished if they were "bad", or rewarded copiously either if they were "good" or if they died killing someone who was "bad". Not everyone agrees on the whole killing thing, though, and again, the experts chime in rightly to point out this looks suspiciously like wishful thinking at best, and a rather manipulative lie at worst.
Self-awareness should not be confused with free will. It should be prima facie obvious that neither one necessarily connotes the other, and better yet, rigorous arguments can be made that humans possess neither trait. After all, how sure are you really that you control your own destiny... let alone know yourself?