A serverless database is defined like any other serverless service: it needs to have high availability, high scalability, and the pricing model must consider its real usage. Satisfying those conditions is particularly hard for databases since performance is a key feature. For a predictable and high performance, databases are usually configured in their own dedicated servers, but serverless requires a shared model to avoid charging the customer for 100% of the time the database is available. In serverless, we want to pay only when a request is done and not when the database is in an idle state.
Currently, only a few services have managed to bring the serverless model to databases. AWS offers just one service: SimpleDB, but it lacks many important features and is extremely limited. For other and better options, you can try FaunaDB, Google...
