What does it mean if a database is ACID compliant?

Context: In computer science, ACID is a set of properties of database transactions intended to guarantee data validity despite errors, power failures, and other mishaps. In the context of databases, a sequence of database operations that satisfies the ACID properties is called a transaction. For example, a transfer of funds from one bank account to another, even involving multiple changes such as debiting one account and crediting another, is a single transaction.

In 1983, Andreas Reuter and Theo Härder coined the acronym ACID, building on earlier work by Jim Gray who named atomicity, consistency, and durability, but not isolation, when characterizing the transaction concept. These four properties are the major guarantees of the transaction paradigm, which has influenced many aspects of development in database systems.

According to Gray and Reuter, the IBM Information Management System supported ACID transactions as early as 1973.

Atomicity
Main article: Atomicity 
Transactions are often composed of multiple statements. Atomicity guarantees that each transaction is treated as a single "unit", which either succeeds completely or fails completely: if any of the statements constituting a transaction fails to complete, the entire transaction fails and the database is left unchanged. An atomic system must guarantee atomicity in each and every situation, including power failures, errors, and crashes. A guarantee of atomicity prevents updates to the database from occurring only partially, which can cause greater problems than rejecting the whole series outright. As a consequence, the transaction cannot be observed to be in progress by another database client. At one moment in time, it has not yet happened, and at the next, it has already occurred in whole.

Consistency
Main article: Consistency 
Consistency ensures that a transaction can only bring the database from one consistent state to another, preserving database invariants: any data written to the database must be valid according to all defined rules, including constraints, cascades, triggers, and any combination thereof. This prevents database corruption by an illegal transaction. Referential integrity guarantees the primary key–foreign key relationship.

Isolation
Main article: Isolation 
Transactions are often executed concurrently. Isolation ensures that concurrent execution of transactions leaves the database in the same state that would have been obtained if the transactions were executed sequentially. Isolation is the main goal of concurrency control; depending on the isolation level used, the effects of an incomplete transaction might not be visible to other transactions.

Durability
Main article: Durability 
Durability guarantees that once a transaction has been committed, it will remain committed even in the case of a system failure. This usually means that completed transactions are recorded in non-volatile memory.

Answer: 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.