On website http://mpgautoparts.com, the subcategories are appearing on the navigation bar. How can I remove them so that it doesn't do it?
When I edit subcategory and select "No" for Include in Navigation Menu, it will remove from the header but also remove it from http://www.mpgautoparts.com/categories/ page too.
I'm not sure why it's removing from both.
Changing that setting to no is the correct thing to do. As for your categories page, it looks custom and if the category dissapears from there too then it must be using the same logic as the menu to get the categories. You would have to change that.
If you don't want to display any category in the main menu, you can also modify the following setting in the configuration:
Catalog -> Category Top Navigation -> Maximal Depth. By setting it to 1, no sub categories should be shown.
If this also affects the display on the category page, you'll have to change something in the implementation of that. As it seems to be custom, we cannot give you any advice on how to do that.