Custom Fields inside Taxonomies?

Hello,

I was just curious,

I’ve been searching and digging into the Post Formats, Custom Post Types, Taxonomies and Custom Fields for a period of time and now i understand how they work and how to use them..

But i was wondering..
All the time i see people add new PRE fixed Widgets fields (Taxonomies) either Category based or Tags based to the admin sidebar?
Now i was thinking will it also be possible to add other options as a Custom Field to the admin page (sidebar)?

Maybe a color-picker or an URL link drop-box for adding an image? maybe a select drop down menu with options?

If so how do i find more about them?
I don’t know the search terms to find them..

Thanks.