Ignacio has explained to you what an MTA is.
Now let's look at the other aspects of your question. The major differences between the various mail servers are
- ancillary functionality
The most commonly used MTAs around the world are sendmail, qmail, postfix, exim and MS Exchange. Another relatively popular MTA (particularly with smaller ISPs) is IMail.
Of these, MS Exchange isn't really an MTA, although it can certainly be used as one. However it offers a lot of functionality that is not essential to an MTA (public folders, calendars, shared task lists and event lists), and it is very popular due to this. However, it doesn't really have a great record when it comes to security and flexibility (although aspects of this have been addressed of late).
Exim is probably the most flexible of all, and it's security is more than adequate for any normal corporate mail server.
Sendmail is considered by many a fossil. It's a good mail server, but a bastard to configure.
Postfix used to be a good choice for a general purpose mail server, but the problem really is that there is just one person looking after the code base, which has reduced its user base considerably over the last few years.
Qmail is probably the most secure of them all.
Any normal MTA will only offer SMTP for communication (and maybe local sockets). Client protocols such as POP3 or IMAP have to be provided by additional software.
Other users of email cannot email you directly. There has to be at least one mail server involved, the one that holds your mailbox. This server will then offer you a variety of choices how to access that mailbox (webmail, IMAP, POP3, depending on your ISP or mail provider). Your mail client (e.g Outlook, Thunderbird, Evolution or whatever you use to read and write email) connects to that mail server to retrieve messages and present them to you.
When you reply to a message (or send a new one), it will be sent to their mail server using SMTP. In most cases (i.e. home users who use an ISP for their email) the message is sent to your ISP's mail server (via SMTP) which will then forward (relay) it to the mail server holding the mailbox for the recipient(s).
If you use email within a company, then (particularly for larger companies) it is not unusual that this company will run their own mail servers. In that case your mail client sends the message to the company's mail server, which then handles the forwarding (relaying).
If you are asking this question because you are generally interested in how this works, then I hope I have explained the basics. If you are asking because you want to set up your own mail server, then my advice to you is: "Don't. Just don't". Email is a complex subject and running a mail server requires time, knowledge and dedication, otherwise you will quickly fall victim to some hacker or spammer, and the end result will be that you can't send or receive email without spending a lot of time getting yourself off the various blacklists.