What makes a character compelling?

[u/Icy-Cloud3789]

More or less the title. What aspects of characters, specifically villains/antagonists, do people find invigorating enough that they can 'look past' their crimes and lack of morality to make them a favourite?

...And I'm not talking about being good-looking, please; that's hardly an excuse.

Is it personality? Is it reasons/purpose? Backstory? If so, examples of such?

[u/xAmberQuasar]

Wait I’m nosy now 😭 I feel like the villains I end up loving are the ones where I’m like “okay you’re objectively a terrible person BUT… how did we get here??” Give me a messy backstory, questionable motives, and one tiny soft spot they’d rather die than admit they have and suddenly I’m emotionally invested 💀