The basic idea for object-oriented database management systems is that an object-oriented program should be able to more-or-less magically save and retrieve any object in a simple way.
Superficially, it should be simple to connect objects in a program to a RDBMS: just make one table for each class of objects.
But in practice, there are serious problems with this: If the object-oriented programming (OOP) uses inheritance, then it is not obvious how to deal with that in the tables.
There are many different object-oriented database systems. This is actually a problem: very few (if any) standards exist, so no system is like another.
Here are some examples of free (GPL license, or similar) object-oriented database systems: