custom comments.php

I’d like some helping customizing the comments section.

I’m currently using my own custom theme and I don’t have a comments.php in my theme so I’m assuming it’s pulling the template from somewhere else. This is how the comments look right now.

https://pristine.nu/review-april-ipsy-bag/#respond

I like the way it looks but I want to customize it more and add to it, however I can’t figure out what I need to do. If I take the comments.php from one of the default themes and add it to my themes folder, it turns into this:

https://24.media.tumblr.com/1bce5397d60c1e15665a95046234d86d/tumblr_n3unb9hD9s1rt59iio9_r2_1280.png

Currently, the style of my comments is customized through my style.css but I’ve read that its functions.php that determines the layout?

What can I add/upload to my themes folder to make my comments layout customizable?