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Atomicity in a database system means that a sequence of events within a transaction must all occur or nothing occurs. Because of this, the transaction cannot be observed by an outside process as “in progress”. It is viewed in one moment as having not occurred yet, and at the next moment as either being complete or having been canceled.
Consistency means that data in a database transaction must be valid according to all defined rules, including any combination of constraints, triggers, and cascades.
Isolation determines how transaction integrity is visible to other users and systems. 1 An example of what this means is when data is still being written, but another user or system wishes to read the data. Lower isolation levels will allow dirty data to be read, while higher isolation levels will prevent the read until the write is complete.
Durability is a property which guarantees that a database transaction which has been committed will survive permanently. An example of this might be a customer purchasing a concert ticket, and right after the transaction completes, the system crashes. Durability in that case ensures that the user definitely has the selected seat. This is accomplished by ensuring that the data has been committed to permanent storage before acknowledging the completion of the transaction.
Some text taken from the links below. I tried to paraphrase as much as possible to explain it in my own words.