ACID is the acronym for the four guarantees a transactional database system provides: Atomicity (a transaction either commits fully or has no effect), Consistency (a transaction takes the database from one valid state to another, respecting all integrity constraints), Isolation (concurrent transactions cannot see each other's in-progress state — they execute as if serialized), and Durability (once committed, a transaction survives crashes and power loss). PostgreSQL, MySQL/InnoDB, MariaDB, SQLite, and most other widely-deployed relational databases are ACID by default.
In a self-hosting context
For self-hosters, ACID is what lets you run a Mattermost or Nextcloud backup as "snapshot the disk" or "pg_dump | gzip | upload" without worrying that you captured a half-applied transaction. Schemaless / NoSQL stores often relax one or more of the four — see Eventual consistency for what that costs you on the read path.