Not clear what do you really want (please see my comment to your question), but most of the shared Web hosting offers include a database management system installed by the hosted provider. You can create one or more databases at the expense of your hard disk quota (which can be formally unlimited), the number of databases can be limited, they are created using one or another "Control Panel" designed for site administration and installed by the hosting provider from the very beginning.
As far as I know, one of the most usual offerings is based on LAMP and MySQL, please see:
http://en.wikipedia.org/wiki/LAMP_%28software_bundle%29[
^],
http://en.wikipedia.org/wiki/MySQL[
^].
There is a number of different options though.
If this is something you are interested in, do your own search:
http://bit.ly/u9DZeH[
^] or maybe
http://bit.ly/xuLjL8[
^].
Remember the main principle here: try before you buy. One good option is free web hosting intended to provide long-term testing followed by upgrade to payed full-featured service. The prices can be quite affordable.
—SA