The skeptic argues that none of our beliefs are really justified, because we have no way of knowing that the world we perceive is the "real" one (e.g. we could be in the Matrix).
The Skeptic's Master Argument:
1. Most of my knowledge about the world depends upon the reliability of my perceptions (e.g. I 'know' I have two hands, because I can see & feel them).
2. It is possible that I could have all the exact same conscious experiences that I'm having now, even if none of them were "true" perceptions (e.g. I could be in the Matrix, and never realise it).
3. Trusting in my perceptions is not justified, unless I can prove (at least beyond reasonable doubt) that they are perceiving reality (i.e. prove that I'm not in the Matrix).
4. I cannot prove beyond reasonable doubt that I'm not in the Matrix.
Therefore, 5. I cannot truly know that I have two hands. (Or, more generally, my beliefs about the world are unjustified.)
"Isn't that a bit improbable?"
The most natural response (the first time you come across the skeptic's argument) is to say "well sure, anything's possible, but it's surely not very LIKELY that my whole life is an illusion!?"
The problem with this response is that it begs the question. There is no way for us to assign probabilities to these rival metaphysical hypotheses. Of course realism SEEMS more likely to us, but this judgement is made based on assumptions we have picked up from our perceptions and everyday life. In other words, realism only seems more likely if you've already assumed realism to be true. If judged objectively, there is no possible evidence which could suggest that realism is ANY more likely than anti-realism, because all our subjective experiences would be exactly the same in either world.
I would answer the skeptic by identifying various different "worlds" that we can talk about, and then asking which of these worlds we are refering to when we claim to have knowledge.
Particularly, I want to distinguish between the "Objectively Real World" (or RW), and the "Common World" (or CW). To explain these terms and make this distinction clearer, consider the scenario portrayed in the Matrix movies. In this case, the RW is that which is ruled by the machines, which humans are blissfully unaware of (forget about Morpheus & co for the moment). The CW, by contrast, is the world inside the Matrix.
It is this matrix-world that humanity have in common, that people talk and argue to each other about. Moreover, I suggest that it is this Common World that is the appropriate domain of knowledge, rather than the external RW outside the matrix. Consider Joe, who just got fired (and knows this fact). One might argue "well, he didn't actually get fired, since none of it is REAL", but I think that rather misses the point.
All knowledge is about some particular world, and purports to represent that world accurately. It thus allows us to explain (& perhaps predict) things about that world. For Joe, who knows he just got fired, which is the relevant world? What events is he trying to explain? What sort of future events is he trying to predict? The answer is evident: he is concerned with the world of his experience, the world where he lives and interacts with other people. In short, he (and his knowledge) is concerned only with the CW, not the RW. As Joe's belief (that he got fired) accurately represents the state of affairs within the CW, we must consider it to be true within the context of the CW. There is no reason to deny that Joe does indeed have knowledge - it is simply knowledge about the Common World, rather than knowledge about the Objective World.
Just in case you doubt that knowledge and truth properly belong within a particular framework, consider another example: Bob is talking to Bertha, and says to her "I had this dream where I was walking down the street..." - and suddenly Bertha interrupts, shouting "Liar! You were lying in bed asleep!"
Bertha's response is quite obviously inappropriate in this case. She is imposing an impractically strict notion of truth, restricting it only to the "real" world, which proves to be a serious (and pointless) obstacle to meaningful communication.
I suggest that the skeptic is making the same mistake. By demanding that our knowledge be about truths of the "Objectively Real World", the skeptic is missing the whole point of knowledge and communication. We are not generally interested in discussing some abstract RW. Instead, normal discussion is concerned with (and takes place within the context of) the Common World which we share and experience. Whether it is "objectively true" or not is irrelevant. Propositions about it can be true within the appropriate context, and that is what matters.
We can have knowledge about the world of our common experience. This crucial and undeniable point is overlooked by the skeptic. In the end, all the skeptic can say is that we cannot know whether the world of our common experience is the "objectively true" world or not. But this is of no great concern, since in our day to day lives, we are not really concerned with some abstract notion of objective truth. Instead, we try to make sense of this world we find ourselves in. No easy task, of course, but not nearly so grim as the skeptic makes out, either.
Update: I notice that Prof. David Chalmers has some similar ideas.