When entering data on a website where does the data go to?

It goes to the websites server space. That is esentially what you are paying for when you buy webspace.

Do you store the data? Where? A hard drive?

You need a database for storing the data.

Does the database need to be on the internet?

Depending on what you need the data for you could just have it email you the contents of the form. If it is something that doesn’t need to be constantly retrieved it would be easier.

You might want to start reading up on PHP and MySQL here.

Usually an encrypted MySQL database.

If you don’t want to build website from scratch, you can use some open source CMS like Wordpress, Joomly, Typo…

Most of them have extensions/plugins for building forms without the need of any programming knowledge (mySQL).

In most cases, it goes to a database such as MySQL which you bind to your PHP code. I suppose you could use indexed flat files, but hardly anyone does it that way these days. If you want to be fancy, you can run your database on a different server than your web server for extra performance if your site becomes popular.