In order to filter by category:

1. Go to the Module Settings
2. Go to the Articles Options page
3. Near the bottom of the page you should see a section called Categories. Check the Filter By Categories checkbox.
4. A checkbox group should appear with all categories. Check the categories you want to be included for the module.
5. You'll also probably want to check the Display Site-Wide Articles checkbox. This will allow the module to display articles from any articles module on the site that meets the selected filters.

The screenshot below highlights the relevant sections:

After re-reading your question, I realized you might want users to be able to filter by categories rather than just doing it through module settings. If that is the case then all you need to do is add the Categories module to the page (usually in the left or right pane). Users can then click on a Category and that will cause the Articles module to filter by that Category.

 Let us know if this or the other solution provided doesn't meet your needs.

I had done all of that except check the All Articles block. Now that I have done that the articles duplicate, or triplicate, but still no filter. I am using DNN 7 - could that be the issue?

This site is DNN7 and it works fine. You can see it working here:
http://www.efficionconsul...om/home/all-articles

It sounds like perhaps there's an odd server caching issue going on. When you click on a category, it should specify the category ID in the page url and then the Articles module will pick that up and filter by it. Without seeing the site, it's hard for me to guess what might be going wrong. Feel free to email me with the site so I can take a look, david [at] efficionconsulting.com