MUA(email client)
Mail User Agent is a 'mail client program', the part you see and use to send and receive mail.
MTA(SMTP server)
Mail Transfer Agent is anything between MUAs. MTAs generally do the routing and transferring of mail between hosts, ISPs, and so on, depending on how detailed one wants to be.
MDA
Mail Delivery Agent. An email arrives and does not need to go further. The MTA gives it to the Mail Delivery Agent, and the MDA puts it in an incoming mailbox. It might call other tools like procmail that then become part of its function.
The MDA accepts a piece of mail from an MTA and performs the actual delivery (including processing aka "filtering") to the user's mail folders
MSA
On an ISP with lots of users that use MUAs on machines that don't have their own MTA, it is not enough to simply let all those users submit their messages to the MTA systems without any authentication. Otherwise spammers will exploit this and abuse the system. Accepting emails from users for sending elsewhere becomes a specialized task, and an MTA that does that is called the MSA.
MRA(POP/IMAP client)
A Mail Retrieval Agent makes connections to a remote mailbox and fetches mail for local use. Fetchmail as a standalone program is a common example.
