What does 大丈夫 Daijoubu mean?

Daijoubu means all right, fine, safe, no problem. It is one of the first words a learner meets and one of the last they fully master, because a second use has grown on top of the first.

Where it comes from

The expression is traced to Chinese, where the components described a fine, sturdy man: 大 is great, and 丈夫 carried the sense of a robust, dependable figure. From a description of a person it broadened into a description of a state of affairs, which is a common enough drift and leaves an odd residue: a word for a strong man now applied to whether a cup is likely to spill.

The everyday use

Asking after someone who has stumbled. Confirming a plan will work. Reassuring a person who is worried. In all of these it means what English all right means, and there is no difficulty.

The use that causes trouble

It has also become a way of declining. Offered a bag at a till, or a second helping, a speaker can answer with this word alone and mean no thank you.

The logic is that the speaker is already fine, so nothing further is required, and the refusal is left implicit. The difficulty is that the same word can equally mean yes, that's fine, go ahead, so the utterance is genuinely ambiguous without tone or gesture. This is not a foreigners' problem: Japanese commentary has discussed the ambiguity, and it is recent enough that speakers of different generations do not always read it identically.

Why English cannot do this

English has the same instinct and lacks the same word. I'm fine and I'm good both work as refusals, and both are recent, and both have irritated people who consider them evasive. But English cannot use fine on its own to decline, because bare fine reads as assent.

So a subtitler faced with a character declining a bag with a single word has to write no thanks, which is correct and loses the softness. The Japanese speaker declined without ever saying no, which was the point of choosing the word.

The pattern underneath

Politeness routinely works by leaving the refusal unsaid and offering a reason instead. English does it with I'm all right, thanks, and Japanese has compressed the same move into one word, which makes it faster to say and harder to interpret.