An object oriented database (OODBMS) is one built using a pure object model, although most experts will accept close approximations of a pure object model. Pure object oriented databases can only be used by pure object-oriented languages such as Smalltalk, or one of its close derivatives. The close approximations to an OODBMS can be accessed by most of the object-oriented languages such as C++. In a pure OO environment, an OODBMS allows the developer to think in purely object oriented terms, making the system more consistent (orthagonal) and often times that makes it more reliable.
A relational database is one built using the relational data model. This is by far the most common kind of database today, built using tools like DB2, MS SQL Server, Oracle, etc. Relational databases are much less expensive to buy than true object-oriented databases, they generally perform much better (on comparable hardware), and it is far easier for me to find relational database expertise than it is to find OODBMS expertise.
A hybrid database is usually an object-oriented framework created to act as an interface between an "impure" object-oriented language like C++ and a relational database manager. The hybrid manager allows the language to access the database as though it were truly object-oriented, while leaving the database itself unchanged. The hybrid design allows the object-oriented programmer to use nearly any OOP feature that they want (much like an OODBMS), while keeping the database itself relational which allows the use of commercially available and supported products, allowing the "best of both worlds" at the cost of the run-time overhead to support the hybrid framework.
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This was first published in November 2002