"what he said". The Mail Transfer Agent (e.g. postfix, exim, etc) is the part of a mailserver that accepts mail, hands it off to a delivery agent if it's local, or hands it off to the recipient's MTA if it's not.

Basically it's the component that handles SMTP. Local delivery, pop3, imap, etc are almost always handed off to another component.

It's usually the difficult component because it's the service that other hosts will interact with. So if my imap service misbehaves, it annoys me until its fixed. If my MTA misbehaves, it's a lasting black mark against my host's reputation.

Message Transfer Agent. Compare with MUA, MDA, etc. Lots of WTF TLAs when trying to build SMTP over TCP/IP, etc. :-)