Solipsism is the belief that one's own mind is the only thing that can be known to exist. It denies the existence of any external reality—everything you experience is simply a product of your own mind. Nobody really believes this, but it's often propped up by philosophers as some sort of exercise in the limitations of evidential arguments—after all, you cannot use evidence to prove that an external reality exists, because by definition (per Solipsism) your mind would be creating said reality.
It's complete bullshit, and contrary to what some philosophers like to profess from their armchairs, it can be shown to be completely and totally false.
Evidential arguments are one way to show that something is false, but they're not the only way. Another way is a conceptual argument. You can demonstrate that something is false by showing it to be conceptually incoherent, nonsensical, or otherwise semantically ambiguous. Regular readers of this blog will no doubt recall that this is my favored approach to demonstrating the non-existence of a god or gods, particularly like that of 'classical theism'—an approach known as theological non-cognitivism or ignosticism.
It's this conceptual approach that we can use to show Solipsism to be false as well. Solipsism says that only one's own mind can be known to exist. Realism, which (in its various forms) holds that the external world does exist, is what allows us to produce a coherent theory of mind in the first place. We can use empirical inquiry to study minds—through cognitive science, neurobiology, and neurocomputation. We know that what we call 'the mind' is connected to physical brain states. We know that the external world not only gives us information through our senses, but is crucial to conceptual reasoning (see embodied metaphor). The study of the mind is a robust science, and it gives us a clear and coherent theory by which we can know our own minds.
Of course on Solipsism, this evidence can't count, because your mind is fabricating it all. But then what is a mind in the first place? How does the mind fabricate all this data? How is new information, or the perception thereof, acquired? It becomes impossible to say what the mind even is, much less how it works. If the mind doesn't exist in reality as we know it, where (and how) does the mind exist? On Solipsism, these questions are unanswerable. It's impossible to develop a coherent theory of mind, and thus the very word 'mind' loses all meaning.
Solipsism is false. No one believes it, but even as an exercise in philosophy, it's stupid.