For data that must be saved and stored correctly, we use PostgreSQL (we usually refer to it as "Postgres").
It's a 30 year old open source database that is highly respected, well supported by documentation and hosting providers, and used by any developer who knows the SQL standard.
In recent years, a movement called NoSQL has gained popularity. Best translated as "not only SQL", tremendous effort has been made to create different kinds of databases for different use cases, often based off academic or industry research.
Our most frequently used NoSQL database is Redis, which we use for storing transient, high quantity read/write data such as activity feeds, tags, background jobs, sessions, tokens, and counters.
Redis is reliable, open-source, and simple. It offers high performance and reliable predictions of its performance. It can flexibly model different data sets but we typically use it for small data structures, not large images, videos, or text documents.
We typically use Heroku Redis to host our production Redis databases.