each Affiliate has his own link. this link is presented to them in their affiliate account, and it's basically a link to your website's root, which also contains the affiliate ID in it
like this:

http://americasnextnaturalmodel.com?atid=1

So, Affiliate A in logged into my site, and requests a new affiliate account. The workflow sends a message to the admin to approve it.
After approval, Affiliate A gets email telling him it is approved.

Affiliate A goes to the website and sees the new affiliate tracker link and that it is approved.
Affiliate A then sends by email to prospective Affiliate B, a message of their own construction, that includes the affiliate tracker link.

Prospective Affiliate B uses the link to register on the website and tracking begins.

Please tell me I am correct.

You are correct... except for one "naming" thing: "Affiliate B" is not really an "affiliate", he is just a "client". Affiliates are those providing the links to the site (like Affiliate A), not those "accessing the site through those links". these are just "clients", referred to the site by an affiliate.

Yes, I understand.

So, if used the atid in a banner ad, i would be tracking the occurrences of its usage and members who registered using it.
As individuals register, the system logs their activity and pairs the atid number with their Username.
Correct?

If there is some other component I want to track, I use your code and insert it.
Can you tell me where the best place this code should be inserted if using Breezingforms for example.

Thanks

yes, correct.

I don't know about Breezingforms.. I'm not familiar with its code... that is something you would have to investigate...