It is quite a debatable topic of what is meant by an Oracle DBA/Architect when he utters the word Oracle. It's a question that is one of the earlier confusions of a novice when he enters in Oracle world and also it's a question that creates a rift in the mind of some seasoned Oracle practitioners as well.
According to Greek Mythology Apollo was termed as the God of fine arts and reflections. To help his followers in the matters of public relations and others Apollo used to speak at its temple. Apollo chose an older woman to convey his messages to common people. The name of that woman was Pythia and she spoke through meaningless cries which were only understood by priests. Those meaningless cries were called Oracle. So every time a person needs any advice from Apollo he would go to the temple and asked for Oracle. The word Oracle means "thing of wisdom".
In modern era Oracle was the name of a project of CIA in early 70s. That project vanished pretty soon and today Oracle is something else. Inspired from Codd's twelve rules for relational model for databases Larry Ellison suggested IBM to build a DBMS that uses those twelve rules. Upon IBM's refusal Larry Ellison and Bob Miner decide to quit their jobs in IBM and work for their own. They hired a room and started to build a DBMS that used relational model of databases. They build their first version in 1980 but did not commercialize it. In 1982 the second version came and they decide to name it Oracle. Today that product is the most popular and most reliable DBMS to ever exist.
It is still amongst the most amazing things in computer world that with all the rapid progress in the software and hardware industry, after almost forty years the basic elements in database design are still Tables, Columns and Rows. It's not because of lack of innovation in database area, it's only because of the fact that those ideas from E.F. Codd were so strong that there never has been a better alternative to that. And the only person to realize the potential of these concepts in late 70s was Larry Ellison.
Now back to the question what's meant by Oracle in today's world? There are three entities that are termed as Oracle and they are Oracle Corporation (Name of the company that Larry Ellison and Bob Miner started in 1978, at that time it was known as Relational Software Inc. Named to Oracle in 1983 and is the second largest software company in the world today.), Oracle Database (The physical set of files that makes storage and retrieval of data possible) and Oracle Server (The RDBMS, the software written in C Language that retrieves and manipulates the data stored in Oracle Database). As you can see that each one of these three entities has their own unique name i.e. Oracle Corporation, Oracle Database and Oracle Server. But for ease and for sometimes for style people rarely use the complete name of all three entities and simply say Oracle. The context on which the name is pronounced determines what is meant by Oracle.