I need to run some code written in Java from my personal website. How do I even begin?

What type of code? Are you making it online (applet), or a launch-able application?

Edit: by what type I mean game or website enhancing program, or a database like code

To execute a java program on your PC you need to install the Java Runtime Environment(JRE).
Go to the Oracle site and find the download for a current version of the jre.

I don't understand what you mean by "from my personal website".
That implies an HTML file with an <APPLET tag. Is that what you have?