My friend showed me a font called "rubber stamp"... I want to make my category names this font. Is this possible? If so, how?

You can use it in HTML as text, but users that don't have that font installed will simply see Times New Roman or another default font.
If you want everyone to see the text in the right font, the only option I know of is making it an image.

Where would it have to be installed in their computer? I would assume the browser they are using becuase that is whats reading the text. I just wanted to make sure.

Second... If I made the text an image using something like photoshop or whatever... how and where would I put what information to make it appear as the category name?

They will need to have it installed on their Operating System.

And you can't dinamically generate these images, so without hacking the PHP files to create the image (have no idea how), you can't use it for the category.

Have you ever personally heard of "rubber stamp" font? I wonder if it's a windows standard font.