The most popular method is mysql as a database engine, which is open source, free, and heavy with documentation. Most hosting services offer a control panel to manipulate the database when you opt to use a database on the site.
IMO, many of these don't give you the full power of database interaction, and from what I've read are often attacked and have vulnerabilities. If you have command line access, god old command-line entry is all that's required.
MySQL documentation [dev.mysql.com]
Basic SQL Tutorials [w3schools.com]
Note that the last is a quick reference to **ALL** SQL languages, not specific to mySQL. The cool thing about mysql is you can download it and install it on your local computer, play with it as required before creating a database on your site and throwing statements at it.
Your form is the browser interface. To interact with the database from the form, you will need some scripting language. The preference these days is PHP, although this can be done with any scripting language: perl, ASP, ColdFusion, etc.
The function of a script is to accept input from your form, then connect to the database to manipulate the data.
PHP tutorials [w3schools.com]
You will find tutorials for other languages there at W3C Schools, once you choose one you can search out other samples and examples on the web.
You could probably do a "quick fix" and download a working script, but if it goes wrong you won't know how to fix it. Get a few tutorials under your belt and you'll be on your way.