MTA stands for Mail Transfer Agent. An MTA is a server-side program which accepts and transmits email messages on behalf of end users. Examples: sendmail, postfix, qmail.
Compare to MUA or Mail User Agent: a program for reading and writing email. Examples: Outlook, Thunderbird, etc.
The terms MTA and MUA were developed in order clarify discussions of email processing. Ambiguous terms such as "email program" could refer to server-side programs or client-side programs.