A digital signature proves that the owner of a private key produced or endorsed a given message. These are used in transactions to prove that an output can be spent in a transaction.
Digital signatures are used in Bitcoin to prove that a transaction is allowed to spend from a given output. Bitcoin uses the Elliptic Curve Digital Signature Algorithm over the secp256k1 curve for producing signatures. The message that a private key signs and is used for validation is the hash of a special version of the transaction.