Aurora is proprietary technology from AWS (not open sourced). PostgreSQL and MySQL are both supported as AuroraDB. Aurora is cloud optimised and claims 5x performance improvement over MySQL on RDS. Aurora storage can automatically grow in increment of 10 GB, up to 128 TB.
Aurora Serverless
Automated database instantiation and auto scaling based on actual usage. Both PostgreSQL and MySQL are supported as Aurora Serverless DB. There is no capacity planning needed and has least management overhead. Serverless is paid per second, and is more cost effective, it is suitable for infrequent, intermittent, or unpredictable workloads.
How does Aurora work
The client will connect to a Proxy Fleet that managed by Aurora, it will automatically instantiate instances when it needs to scale up or down. These Aurora volume is sharing storage.
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Reference* - Udemy Ultimate AWS Certified Cloud Practitioner CLF-C02