Amazon Aurora is a fully managed relational database service from Amazon Web Services, fully compatible with PostgreSQL and MySQL. AWS describes it as a serverless database with zero infrastructure management, including serverless options, and says it offers multi-Region availability. Applications written for PostgreSQL or MySQL can use it as a managed service in place of running those engines on their own servers, with AWS handling the underlying infrastructure.
Amazon Web Services is a subsidiary of Amazon, a US group. Aurora is delivered as a managed AWS service, so the provider's jurisdiction and access controls apply to the data it holds. The AWS GDPR Center states that customers decide where their customer data is stored, including the geographic region, that AWS protects customer data regardless of the region chosen, and that customers can use AWS services to process personal data covered by the GDPR. AWS has Regions in Europe, but that page does not list them, so a European organisation must select and verify an EU Region. Transfers from the EEA are covered by standard contractual clauses.