RDBMS stands for Relational Database Management System. An RDBMS is a program that allows you to create and administer a relational database.
A relational database (RDB) is a particular type of database which stores data in tables that can be related to each other. Most databases used for modern programs and websites are RDBs. An RDB management system allows developers to maintain and interact with their relational database. While non-relational DBMS applications store data as files, an RDBMS stores data in table form. This tabular design means that relationships can be defined between tables, which was not possible with non-relational databases. Each table is made up of rows and columns. One or more columns within the table may be identified as the "Primary Key". This key can be used to define a relationship to another table that has the same field(s). While a key is not required for a relationship to be used, keys can both enforce data integrity and can be used to optimize database performance, making RDBs the most commonly used database design today. The original relational database design was developed by Edgar Codd in 1970. Designs have evolved considerably since then, and while most modern databases don't conform to all aspects of his original design, they can still be considered relational databases. Almost all modern RDBMSs use the language SQL (Structured Query Language) to access and manipulate the data they contain.
While Microsoft began developing an RDBMS offering in the mid-1990s, their release of SQL Server 2000 marked the company's true entry into the RDBMS market. While Microsoft's product is most at home on Windows systems, a restriction that prevents some from using it, Microsoft SQL Server is one of the most robust RDBMSs on the market. It offers a user-friendly GUI and most licenses come with a suite of advanced analytical software. The latest full release is Microsoft SQL Server 2014, and the upcoming 2016 edition is now available to preview.
Oracle first released its RDBMS in 1980, and it continues to be a very strong player in the market today. Oracle's RDBMS is a secure and full-featured system, with support on multiple operating systems. A wide range of first- and third-party GUIs are also available for users to choose from. The current edition is Oracle Database 12c.
IBM's DB2 (Short for "Database 2") was a market leader for years, but has fallen behind more recently. DB2 is highly scale-able, with editions and licenses available for every need, from single-user to enterprise systems. The most popular is the 'Mainframe' edition for z/OS. DB2's most recent release is 10.5, and is available for Linux, UNIX and Windows.
MySQL is by far the most popular open-source database. It is highly scale-able, and used to power the largest websites. MySQL can run on a range of operating systems, including Windows, Linux, and OSX. Being an open source product, tools and support can be very hit or miss, but its dedicated fans have made some excellent tools available. As an open source product, it's an especially good choice for small projects where cost is a concern. The current edition of MySQL is 5.7.