According to Gallup's famous "mirror test", you can test whether an animal is self-aware by whether it can learn to recognize itself in a mirror. In particular, if you secretly mark the animal's forehead, and it sees this mark in the mirror and responds by touching its own forehead, then this shows that the animal recognizes itself in the mirror. So, the argument goes, the animal must have a concept of self, and be self-aware. But despite the initial plausibility, this conclusion doesn't follow at all.
All living organisms draw some sort of distinction between self and other. For example, the job of our immune system is to identify and destroy alien cells, whilst leaving our own ones untouched. But no-one would thereupon conclude that our immune system is self-aware. Building on this sort of idea, our lecturer pointed out what I think is a devastating criticism of Gallup's test.
Suppose experimenters marked the animal's arm instead, in a clearly visible spot. Now, we surely wouldn't be surprised if the animal then touched the mark on its arm. It can identify its own arm, but that doesn't entail full-blown self-awareness. But why does bringing a mirror into the picture make any difference? A mirror is merely a perceptual tool - it allows you to see things that you might not otherwise see. In particular, it allows an animal to see its own forehead, in addition to its arms and such. Most animals can't make use of this -- they can't work out that the mirror gives them a view of themselves (they either think the image is another animal, or else ignore it entirely). Some apes are more intelligent in this sense -- they can make use of the mirror as a perceptual tool. But it still doesn't imply anything new about psychological self-awareness.
What we really want to know is whether animals know that they have minds. Do they have a concept of self (as opposed to simply being able to recognize themselves)? Can they think about their own beliefs and desires, recognizing that they have thoughts, and that they have a life, and are the same creature today that they were yesterday? These are the important questions, but they're questions that the mirror test is silent on. Despite all the hype, it doesn't really say anything interesting about self-awareness at all.