put my blog content somewhere else

Hi, I think probably this isn’t the place to ask this, but still, I’m trying to do something like this: https://rhizome.org/widget/
That is, some widget to show one or two of my blog categories so people an include that in their posts. I ‘ve searched everywhere, I know it is done with some php and javascript, but haven’t found the code for it.
I’m not looking for a ready made widget nor for a plugin, I just want some code I can tweak and make it work with my blog.
Any help greatly appreciated since I’m clueless even on what is the name for such a code. Thanks a lot!
Indira