Solipsism is the view that only your own mind is certain to exist. Other people might be philosophical zombies — outwardly indistinguishable from the conscious, but with nothing going on inside. The position sounds obviously crazy, but it's surprisingly hard to refute on its own terms.
You can't introspect another person's experience. All you have access to is behavior — what they do, what they say, how they react. You infer minds behind the behavior because behavior matches what you'd do if you had a mind. But the inference is exactly the kind of inductive leap Hume warned about: just because the pattern has held so far doesn't prove it generalizes.
Most philosophers don't take solipsism as a serious metaphysical position. They take it as a useful boundary case that reveals something about the limits of proof. Wittgenstein argued that the very meaning of "mind" or "pain" depends on a community of users; a private language doesn't make sense, so radical solipsism is incoherent on its own terms. Levinas turned the problem upside down: the encounter with the *face* of the other is the foundational moral fact, prior to any inferred metaphysics. Whether you find any of these convincing is itself a test of how seriously you take the question.