A proprietary Java-like programming language for the Force.com Platform, not to be confused with Oracle's Application Express (oracle-apex).
From the Salesforce.com page What is Apex?:
Apex is a strongly typed, object-oriented programming language that allows developers to execute flow and transaction control statements on the Force.com platform server in conjunction with calls to the Force.com API. Using syntax that looks like Java and acts like database stored procedures, Apex enables developers to add business logic to most system events, including button clicks, related record updates, and Visualforce pages. Apex code can be initiated by Web service requests and from triggers on objects.
For questions that are related to this tag you can use shorter tag - apex.
(Please use oracle-apex to refer to Oracle Application Express.)
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