Digital Signature
Proof, produced with a private key, that the holder of that key approved one specific message such as a transaction.
The signer computes a signature over the hash of the exact message, and anyone can verify it against the public key, so a signature authorizing one amount and recipient cannot be reused for a different amount or recipient. Nodes reject transactions whose signatures do not verify, which is the entire authorization system of a public blockchain: there are no accounts to log into and no operator to appeal to. Signatures are also used off-chain, for signing into applications and for approvals such as permit-style token allowances, where a signed message lets a contract move tokens later without a transaction being visible at the moment of signing. That last pattern is why wallets warn about signature requests, and it is behind a large share of consumer losses.
In practice
Signing a permit message grants a contract permission to transfer a token balance later, so the loss occurs at a moment the victim did not associate with a transaction.
The common misunderstanding
Signing a message is not always harmless; some off-chain signatures authorize a contract to move your tokens afterwards, with nothing appearing on chain at the time.