How could I execute a line of terminal code, like this
java -Xmx1024M -Xms1024M -jar minecraft_server.jar nogui
from the press of a button on a website hosted on the same machine (with apache).
You could achieve this in several ways:
Write a CGI script which gets run when you visit a certain web page. The CGI script would then start the shell command. Read up on CGI in the Apache httpd documentation.
Put a PHP page on your server, and in that page put a system() call executing the program you wish to execute. Any PHP tutorial will show you how to do this.
However, if you intend put this page on the public internet, you can be quite sure this will be abused. For one, make sure the program can only start once or your server will be on its knees on day 1.
