MySQL [url=http://www.mysql.com][/url] is the self-proclaimed 'The world's most popular open source database'. It's the standard database system you get for a web hosting account.
Unlimited means that we put no defined limit on the number of those databases you may use (as long as it stays reasonable and it doesn't create problems). A lot of web hosts put restrictions on the number of databases and let you pay/upgrade if you need more of them. The problem with that is that with a lit of modifying, you can let several websites/software, use the same database. So actually you only need one, but for your convenience you get many.