What is a database?
A database is a way to store information that can be retrieved and displayed on your website.
A database is made up of tables in which detailed information can be stored and retrieved; for example, consider a day-planner, with the hours of the day from top to bottom on the left, the days of the week along the top, and empty spaces between for you to write in appointments or things to do. This is a paper version of a database table.
If you have subscribers for an email newsletter, a database solution would allow you to store their information. Or maybe you want to allow your visitors to comment on your site. A database would allow you store those comments and let you read them for appropriate content before allowing them to be seen on your site.