A transcriber is someone who writes down what someone else is saying. In court, there's always a transcriber who makes a record of the testimony.
If you write down everything someone says, you're transcribing their words. Someone who does this as a job — or for another reason — is a transcriber. Transcribers can use shorthand, though more often they type. You have to be a fast typist to be a transcriber because people speak very fast. Transcribers are often used in legal contexts, either in court or when someone is making a statement elsewhere for the record. Journalists often have to be transcribers while conducting interviews.