The first game I ever remember playing on my computer was the 1983 game by Windmill Software, Digger. This game had you controlling a little digging machine, similar in appearance to a bulldozer as you navigated your way around several levels trying to collect all the gems and gold so that not only could you progress to the next level, but get a higher score too. To impede your progress, you would be chased by monsters called hobbins and nobbins. You could either continue to evade them or fight back with one of three ways: (1) Fire a cannon from your Digger (although it takes a few seconds to reload), (2) dislodge bags of gold so that they fall on top of your enemies or (3) wait until a cherry/bonus appears and after consuming it you'll be able to eat the enemies, for a limited time (a bit like Pac-Man). Why the game was so memorable for me was probably for its fun gameplay, bright graphics (red, orange and green CGA) and music (it had Popcorn as its main theme and the William Tell Overture when you got the bonus).
The developers for this game, Windmill Software actually still exist in one form or another, however, they don't produce computer games anymore. Jo-Anne Kempe (ex-wife of game producer, Rob Sleath) now runs the Canadian company which is involved in developing management information systems. Rob Sleath is now an IT Consultant/Webpage developer.
If you're wanting to still play Digger today, thanks to Andrew Jenner, you can download a remade version for Windows, free of charge.