I'm new to mail servers and still trying to understand all the components.
What I read is that there are many MTAs, but the 4 common ones are qmail, Postfix, Sendmail, and Exim.
I also found this list of mail servers. http://en.wikipedia.org/wiki/List_of_mail_servers
What's confusing me is that mail servers like Zimbra and Atmail are listed at the same level as qmail/Postfix, etc. Are mail servers and MTAs the same thing? I thought (correct me if wrong) that a mail server includes an MTA as one of its components.
The problem you're having is that "mail server" is an imprecise term. Some people take it to mean "MTA" (because that's all you need to have a server that handles mail), whilst others take it to mean a server that receives mail and stores it for users to manipulate, while still other people have other definitions entirely.
The four software packages you list are, indeed, MTAs, and have little-to-no other functionality provided. Zimbra, et al are what I would call something like "mail service suites", but really there isn't any fully-accepted terminology. It's best to do your own investigation into what a particular program, package or suite does rather than try to rely on imprecise terminology to gauge suitability. (See also: "cloud").