A popular open-source embeddable (i.e. non client-server) RDBMS implemented as a C library.
SQLite is a C library that manages an in-memory or persistent database structure. It does not have a separate server, but links directly into the application. Many language bindings for SQLite are available, The Wikipedia entry has a list under the 'Features' section.
SQLite is used by a variety of commercial and open-source applications, and is developed by the SQLite consortium. The system is donated to the public domain although the consortium will grant explicit licenses.