This question got me thinking, and I now realize that I don't know anything about the internals of MTAs.
What exactly does an MTA do? Everything after the SMTP protocol seems like dark magic to me. Let's say that I wanted to code a minimalistic MTA (or MDA) just for sending emails, what would I need to learn/do?
Edit: I don't actually plan on writing an MTA, I just want to understand how it works internally.
--- edit after somehow noticing you talked about possibly writing a MTA ---
To write a MTA, you need to open a server socket. When someone connects, you need to send and receive text (ascii) data on that socket in compliance with the SMTP protocol. SMTP is very chatty, so you can expect a few rounds of communication.
The initial round of communication typically tells you whether SMTP is supported or ESMTP is supported. The second (optional) round of communication is to determine security / encryption / feature support. Eventually the "client" side will ask to send a message to a particular address / set of addresses. When done, the server will indicate that it's ready to get the body of the email message. When the body of the message (and it's optinal attachments) have all been transmitted, the MTA will tell you it received the message fine. At that point in time, the MTA will act as a client to other MTAs discovered via DNS MX records to get your email closer to it's destination MTA which will copy it into someone's inbox.
So an MTA is needed because mail delivery on the client side is the equivalent to handing a physical letter to a post office. Post offices are responsible for inter-postoffice routing (which parallels to MTA-to-MTA transmission). The destination Post office is then responsible for delivery of the letter to the post office box or home address (which parallels one's computer inbox).
They don't call it e-mail for nothing.
--- original post follows --- A MTA will accept a mail message, see if it can forward or deliver it, respond if it can be forwarded or delivered, and then forward or deliver it if it indicated it could.
How the message gets closer to it's final destination usually has a bit to do with DNS. MX (mail exchange) records in DNS indicate servers which are responsible (or at least closer to the responsible server) for particular email domain names. It is not possible to fully understand how a mail message gets closer to it's destination without understanding how DNS works.
A MTA typically looks at the delivery address, and either is configured to be the "end point" of the email address's mail domain, or knows that server XYZ is one hop closer to the email address's mail domain. If it's an endpoint, it will copy the message from the wire into someone's inbox. If it's relaying it will "forward" the message to the next MTA.