Consider what happens when you defend a judgement of beauty. You do not say "I like it" and stop; you point at things — the way a line resolves, how a phrase turns, the balance of a composition. You offer reasons, and you expect them to land. That behaviour is odd if beauty were merely a preference, and it is the datum every theory has to explain.
Plato took the strongest realist line. Beautiful objects are beautiful by participating in the Form of Beauty, which is perfect, unchanging, and not itself any particular beautiful thing. On this account aesthetic education is a kind of ascent: from beautiful bodies to beautiful practices to beauty itself. The attraction is that it explains why judgements feel like discoveries. The difficulty is that it requires a metaphysics many find hard to accept, and it struggles with the sheer variability of what people have found beautiful.
Hume moved beauty into the observer, and then did the interesting work of preventing that from collapsing into relativism. Beauty, he wrote, is no quality in things themselves; it exists in the mind that contemplates them. But he immediately noticed that we do not treat all judgements as equal — anyone ranking a trivial writer above a great one would be dismissed, and rightly. His solution was the qualified critic: someone with practice, wide comparison, freedom from prejudice, and delicate discrimination. The joint verdict of such critics, sustained over time, constitutes the standard of taste. It is a subjective account that recovers most of what objectivity was wanted for.
Kant's account is the subtlest and the most influential. A judgement of beauty, he argued, is disinterested — you are not judging whether the thing is useful to you, or whether you want to possess it, but responding to it for its own sake. It rests on a feeling rather than on applying a concept, which makes it subjective. Yet when we say something is beautiful we are not reporting a private quirk; we speak as though others ought to agree. Kant explains this by locating the response in the free play of imagination and understanding — capacities every rational being has — so the judgement is subjective in origin but universal in claim.
A final possibility deserves mention: perhaps the question is badly formed. "Beautiful" may cover several distinct things — formal elegance, emotional power, technical mastery, moral seriousness, sheer novelty — and if so, the interminability of the debate would be explained by our having asked about one property when there were always five. That would not dissolve the disagreement, but it would relocate it somewhere more tractable.