I have healthydogcareonline.com and I want to submit that to a site directory. would I also have to submite healthydogcareonline.com/blog too?
You can submit both. But in many cases directory owners does not like two links from same domain. They may even delete both if you submit multiple urls. So I would suggest stick with one.
You can only submit the blog one in directories that accept deep links. Most directories only allow you to submit the main domain. You may be able to get away with it if it can be accessed as blog.domain. com
If your blog accepted its even better. Deep links are considered better as backlinks. Many webmasters however does not accept deep links.
when people advertise directory submission, is link directories or search engine submissions. cause I need both...
Search engine submissions aren't really necessary. Search engines use crawlers for that. Directory submissions help you get links on other sites, doing that attempts to establish that your site is popular so other people are linking to yours. However getting links from relevant sites brings best results.