Adding custom images to pages

Greetings good folks! Please excuse my lack of knowledge…

I’m trying to add a conditional icon to my homepage/categories/search results pages inside the post boxes. For example, let’s say I have a Videos category. Any posts on the above pages that are tagged to this category I would like to display an image I will upload for this category (let’s just say a youtube icon).

It would need to contain an if function such that the icon only appeared if it fit that category. Some posts would have multiple categories and thus multiple icons. I would like these icons to be bottom and right aligned/floated.

Again, not talking about putting anything in the posts themselves, just wanting the icons to show in the post boxes on the homepage/categories/search results pages.

Any help with how to do this would be greatly appreciated.