Spacewar! – This was probably the first video game I remember playing. The link here shows the orginal game before it was turned into a quarter sucking fest. A link at the site has a java applet that plays the game too.
I used to play this with my grandfather whenever we went to the mall for arcade games. Each quarter upped the time limit to the match, if I remember correctly. It even had a ship that looked like the Asteroid’s ship and another ship that looks like the Enterprise.
I’ve yet to see a remake that emulates it like the arcade. Nothing has come too close yet. (sounds, parts of the ship blowing off, etc.) The arcade version let you blow up part of the ship and still play (ie. an engine could be blown up, but you could still fire.) Not sure if the above applet has it because it’s hard to play alone before the other ship gets sucked into the sun.