Not allowed by law. In plain meaning it is the same as 'illegal', but the two words live in different rooms, and that is the part worth learning.

'Illegal' is what ordinary people say: illegal parking, illegal drugs, it's illegal to do that. 'Unlawful' is the word of lawyers, police reports, contracts, and court judgments: 'unlawful killing', 'unlawful assembly', 'unlawful dismissal', 'entering the premises unlawfully'. If you hear 'unlawful' on the news, you are usually hearing someone quote an official document. So a Taiwanese student writing an essay or an email should normally choose 'illegal'; choose 'unlawful' when you are writing in a legal register or naming a fixed legal phrase.

There is also a real difference lawyers make, though it is not absolute. 'Illegal' tends to mean 'there is a law forbidding this, and breaking it is a crime'. 'Unlawful' is wider and softer: it can simply mean 'the law does not support this' — the act may be invalid, or may make you liable to be sued, without anyone being a criminal. That is why a court says a killing was 'unlawful' when it does not want to name a particular crime, and why an employer's decision can be 'unlawful' when nobody goes to prison for it.

One classic phrase to know: in English inquests, 'death by unlawful killing' is a formal verdict. And note the near-twin 'lawless', which is a different word entirely — it describes a place or person with no law at all, not one act against the law.