How to create archives if my theme doesn't have archive files?

I have a theme that doesn’t have an archive.php or a category.php. I’ve set up categories for my blog and each post is assigned a category, but when I go to the category page, it only shows an exact replica of the blog.

I tried creating a category.php page but nothing in this file has any effect whatsoever on the actual category page.

Any thoughts on how I can make my theme pull the information for the category pages from category.php? (instead of index.php which it is currently doing)
This is my blog. Click the post categories to see… nothing happen.