What does it mean if a database is ACID compliant?

ACID is an acronym that stands for Atomicity, Consistency, Isolation, and Durability.
Atomicity means that actions occur either entirely or not at all; there is no partial write or situation where only part of an action was completed. For a database, this means that if data was written in a single transaction, that transaction either completes entirely or fails completely with no data left behind in a partial or corrupt state.
Consistency means that transactions take the state from one valid state to another valid state, with no way to enter an invalid state.
Isolation means that multiple readers/writers to the same database do not interfere with each other and can proceed without worrying about the effects of another action occurring.
Durability means that state is preserved across failures and shutdowns, e.g. that state is persisted to permanent storage.