A database is a collection of data, most commonly arranged in tables having rows and columns.
A database program is a program for sorting, storing, and manipulation of large amounts of data such that the user can get the information she needs quickly. One would use a database for example to store all the names and addresses of the voting population of Los Angeles for example.
Relational databases have become popular due to high speed. There are some big database vendors in the software business such as Oracle, IBM's DB2, SAP, SYBASE, Microsoft SQL Server, Versant as well as some free software alternatives such as postgresql, firebird, and MySQL.
SQL created by EF Codd at IBM (see his paper the relational model) is Structured Query Language, a language for manipulating large data tables within a database. Most relational databases have SQL embedded within them. Users ask for the data they write a "query" using SQL and the database quickly returns the data. Versant is an example of an object oriented database. OODB-s offer a different strategy than relational db-s that some claim can lead to much greater speed. The market now seems to favor relational databases currently and only time will tell if OODB-s catch on.