Will I always need a CMS with a shopping cart platform?
No, but I would recommend one in most cases. A Content Management System allows you to add content (articles, web pages, etc) to your site as a user or admin. Without a CMS, users cannot add content to your site and admins would have to know HTML in order to create web pages.
Popular CMS solutions run millions of websites each, so you can develop high quality websites faster, with more features and consistency. Without a CMS you are limiting your site's content and also who can add that content.
Which is an example of an external one?
Go to this site: http://www.nationwidebarcode.com/ean-upc-barcodes/
In the right side bar you will see an "Add To Cart" button. Push it. You are instantly taken off the "nationwidebarcode" site and brought to the "e-junkie" site to complete the transaction. Some "shopping cart" solutions only work in this manner.
Accepting payments is a 3 part process:
1. Shopping cart. This is where a user designates what items he or she wishes to purchase. The items are totaled along with any tax and shipping charges.
2. Payment gateway. This is basically the bank authorization. They approve the purchase. These will be secure pages which collect credit card or other banking information.
3. Merchant Account. This is your account which will receive the money.
The above is my understanding of the process. As I shared, this is simply knowledge I have collected and I do not consider myself an expert in this area. If anyone wishes to add their knowledge it would be appreciated.
Does the shopping cart control that or the cms system?
The CMS controls the pages created by the CMS and the shopping cart controls the pages it creates. These are two different systems co-existing on the same site.
Some shopping carts come with built in SEO features. Others do not. Some integrate well with your CMS allowing the CMS to overwrite the default SEO settings.
I really think you would be best served by hiring a developer to build your site. I would recommend checking with friends and other business owners to locate a reliable company. There is a huge range of knowledge, experience and prices. Take your time and locate a developer who will create a site based on your needs.