Sendmail and QMail, but also others like Postfix (a sendmail look-alike), are called MTAs (Mail Transport Agents). They are designed, as their name suggest, to transport mails over a network.
In the case of sendmail, you can also use it directly from the command line to send out mails, as if it was a MUA (Mail User Agent).
Basically, a MUA is a piece of software that acts as a mail client. It allows you to read mails and send them out. Most MUAs will simply initiate a connection to an MTA for sending out the mails. The MTA is considered responsible for the correct routing of the mail (and for performing some mail filtering).
On most hosts, an MTA like QMail or Sendmail is running and listening on port 25 (SMTP - Simple Mail Transfer Protocol). Any program that connects to this port and then issues commands following the SMTP protocol, will send out mails via the MTA. This is handy for sending out mails. Those MTAs are only listening to port 25 on the "localhost" loopback network interface (by default, for security reasons), so they cannot be used from other hosts than the one they are running on.
A "mail server", as you call it, however, is defined as a machine that:
-accepts incoming mails (again, port 25) from other machines
-allows the retrieval of mails that were received (ie via POP3 or IMAP protocols typically).
Since most MTAs on simple hosts do not accept mails from other machines, they are not to be considered as "complete mail servers".
Most MUAs know how to contact mail servers via SMTP to send out mails and via POP3/IMAP to retrieve mails.
Personnally, I don't know Exim, but on the Exim home page - http://www.exim.org/
-, the authors of Exim claim
that it is an MTA, just like sendmail and QMail.