It comes in two varieties: the belief that the material world is entirely imaginary, and the belief that the material world is real but that I alone am conscious, everyone else being a robot, a zombie or whatever. There could also be said to a third variety in the Cartesian-style method of doubt, where solipsism is assumed as a way of starting to fill in the rest of what one can be sure about, which Descartes unfortunately does via the ontological argument.
It might at first be thought that solipsism collapses completely because of analogy - other people behave as I do and I have a mind, therefore they are conscious too. However, this fails because any of us only has a single example of a conscious mind, which is not even enough to base an inductive argument on.
Solipsism is also one extreme of scepticism whose other end is behaviourism - the doctrine that mental states "just are" behaviour and that there is no internal private world inaccessible to others. Both of these positions look about equally absurd to me, but there may be some mileage in looking at the life of the mind as open, as with empathy. A feminist argument against solipsism is that women can feel at one with others and men can't, but this seems like a position which can't be tested to me although i have some interest in the idea that some people may have that kind of access. This also brings up the issue of the autistic spectrum - Asperger's and autism itself - which however, i feel is more to do with salience than theory of mind as such. Another approach is that there are simply no minds, or that all minds are one.
Jean-Paul Sartre, like continental philosophers generally, sees the very idea of the problem of other minds as a kind of scandal which ignores our authentic experience of being an object in the mind of another due to shame. He expresses this in his concept of 'The Look'. Other continental philosophers are also likely to "extrude thoughts from the mind" and see consciousness as dependent upon community, relationships or language or other system of signs, and therefore the problem of other minds is likely to look like a pseudo-problem to them.
Finally, belief in other minds can be seen as a theory which is the best working explanation of the situation as we see it. We can't prove or even corroborate it, but it's a good working model of reality.
Interestingly, most people seem to opt for the "life could be a dream" version of solipsism. That is, they don't actually believe in it, but the one they think of is the one where the world seems to be imaginary, not the one where all other beings are zombies and the world may as well be as it's perceived otherwise. I only realised this today. I find the other one more interesting. There's also a kind of fake solipsism where everybody else is dead for some reason, as in 'Red Dwarf', or perhaps 'The Quiet Earth' (which can be downloaded here! Wow!) In fact, 'Dave Colling, Space Cade' or whatever it's called, would seem to be more feasible as 'Red Dwarf' at least has the Cat, but 'Dave Collins' has aliens, so neither are really solipsistic. 'I Am Legend' (the book - i haven't seen the film) and zombie movies presumably are also quasi-solipsistic or at least the threat exists there.
I might be able to make the Turing video tomorrow. Wait and see.