Idealism
Idealism holds that mind is not one more furnishing of the world but the thing the world is made of, or at any rate the thing beyond which we cannot get.[1] It is easy to caricature as "nothing is real", and that is not the claim. The idealist agrees the table is solid, public, and there tomorrow; the disagreement with realism is about what that consists in.
Two versions are worth separating, because they are argued for in completely different ways. Berkeley's version takes the empiricist premise seriously: everything you have ever had access to is a perception, so a thing's existence is nothing over and above being perceived — esse est percipi. The table persists unperceived by you because it is perceived by God, which is the system's cost, and its notorious feature. Kant's transcendental idealism claims something weaker and stranger: space, time, and causal structure are the forms our minds impose on experience, so we know the world as it appears — phenomena — and never the thing-in-itself.[2] Kant is emphatic that this is compatible with empirical realism: within experience, tables are perfectly objective.
The tempting refutation is Samuel Johnson's — kick a stone, "I refute it thus". It misses, because the idealist predicts the kick and the pain exactly as the realist does; what is in dispute is what makes those experiences true, not whether they occur.
The serious pressure comes from elsewhere. Against Berkeley: perceptions have a regularity and an independence from our will that cries out for explanation, and "God sustains them" explains it only by adding an enormous commitment. Against Kant: a thing-in-itself that causes our experiences while being outside the categories — including causation — looks like it violates its own rules, an objection raised in his lifetime and never fully settled.
What keeps idealism alive is the difficulty of the opposite direction: no one has produced an argument from experience to a mind-independent world that doesn't already assume one. That gap is the problem of the external world.