Oracle is the most popular relational database. It runs on both Unix and Windows. It used to be many times more expensive than SQL Server and DB2, but it has come down a lot in price.
SQL Server is Microsoft's database and, not surprisingly, only runs on Windows. It has only a slightly higher market share than Oracle on Windows machines. Many people find it easier to use than Oracle.
IBM's DB2 was one of the earliest players in the database market. It is still very commonly used on mainframes and runs on both Windows and Unix.
Because of its small size, its speediness, and its very good documentation, MySQL has quickly become the most popular open source database. MySQL is available on both Windows and Unix, but it lacks some key features such as support for stored procedures.
Until recently, PostgreSQL was the most popular open source database until that spot was taken over by MySQL. PostgreSQL now calls itself "the world's most advanced Open Source database software." It is certainly a featureful and robust database management system and a good choice for people who want some of the advanced features that MySQL doesn't yet have.