Numerous Web hosting plans offer database services of one sort or another. There may be a small extra charge (e.g., $2.25 per month).
Calls to the database engine are typically either by CGI scripts or by (usually proprietary) scripting language in the Web pages. A product such as ColdFusion can do a lot to automate defining the connection and interface.
An alternative, if immediate updates are not essential, is to generate static HTML pages from a database that need not reside on the server. This may be a good idea not only because it is easier to implement, but also because it tends to put less strain on the server.
Many different products are available. For example, PHP is a free (Open Source) server-side HTML-embedded scripting language including various database access functions. MySQL is a popular choice for a database backend on Linux machines; APIs are available to allow it to be used with Java or Perl.