The method
Connect to port 80 on the host and send it
HEAD / HTTP/1.0
This needs to be followed by carriage-return + line-feed twice
You'll get back something like this
HTTP/1.1 200 OK
Date: Fri, 03 Oct 2008 12:39:43 GMT
Server: Apache/2.2.9 (Ubuntu) DAV/2 SVN/1.5.0 PHP/5.2.6-1ubuntu4 with Suhosin-Patch mod_perl/2.0.4 Perl/v5.10.0
Last-Modified: Thu, 02 Aug 2007 20:50:09 GMT
ETag: "438118-197-436bd96872240"
Accept-Ranges: bytes
Content-Length: 407
Connection: close
Content-Type: text/html; charset=UTF-8
You can then extract the apache version from the Server: header
Typical tools you can use
You could use the HEAD utility which comes with a full install of Perl's LWP library, e.g.
HEAD http://your.webserver.com/
Or, use the curl utility, e.g.
curl --head http://your.webserver.com/
You could also use a browser extension which lets you view server headers, such as Live HTTP Headers or Firebug for Firefox, or Fiddler for IE
Stuck with Windows?
Finally. if you're on Windows, and have nothing else at your disposal, open a command prompt (Start Menu->Run, type "cmd" and press return), and then type this
telnet your.webserver.com 80
Then type (carefully, your characters won't be echoed back)
HEAD / HTTP/1.0
Press return twice and you'll see the server headers.
Other methods
As mentioned by cfeduke and Veynom, the server may be set to return limited information in the Server: header. Try and upload a PHP script to your host with this in it
<?php phpinfo() ?>
Request the page with a web browser and you should see the Apache version reported there.
You could also try and use PHPShell to have a poke around, try a command like
/usr/sbin/apache2 -V