Can I write something on the screen with Apple Pencil?

Perhaps these are the resources that you are looking for - in particular the iPad User Guide and details of the Scribble feature:

Enter text with Scribble on iPad - Apple Support

Use Apple Pencil with your iPad - Apple Support

Write something directly on the screen? Apple Pencil is not a real pencil or pen, so no, you cannot 'write directly on the screen'

You can't write on your Home Screen, if that's what you are asking.

For most things, you can use an Apple Pencil to replace your finger for tapping/selecting things on the display.. otherwise, in apps that support it, you could 'draw' with that pencil. For example, Notes, which is discussed in the article you came here from.. you can create drawings. Or in a PDF that requires a signature.. you can use the Apple Pencil to sign it rather than printing a copy, physically signing with a real pen, scanning and emailing back to someone.