A
database management system, or
DBMS, is a piece of
software which allows you to create and access
databases. Most usually, a DBMS follows a
client/server model: the
server component, the DBMS program itself, manages the data on
disk and accepts connections from
clients which instruct it to store,
query, or modify data. Popular DBMS packages include
PostgreSQL,
mySQL,
Microsoft SQL Server,
Microsoft Access,
Informix, and
Oracle.
Most of the well-reputed DBMS software today falls into the relational database management system, or RDBMS, category. An RDBMS is any DBMS that lets you create relational databases -- databases whose behavior is mathematically defined by certain axioms based on set theory. PostgreSQL, Microsoft SQL Server, Informix, and Oracle are all RDBMSes. Microsoft Access and FileMaker Pro are not relational. mySQL borders on being relational.
Most modern DBMSes -- including most of those mentioned above -- speak some dialect of IBM's Structured Query Language, or SQL.