I'm comforted that so many people have mentioned games that came out around 1984. It makes me feel less old.
I think the first game I ever played was on the BBC Micro at school. No idea what I played first. There was a castle building simulator that, as far as we could tell, was impossible to win at because no matter how you assigned your "workers" and "supervisors" there were never enough people to get it finished before the barbarians showed up. There was a typing tutor with a wee man climbing stairs of ever-longer words. That one was actually fun. There was a zoo keeping game where you had to move the animals from cage to cage without them clashing- it was basically a sliding block puzzle with the chance of getting eaten. I have vague memories of a text adventure where I don't think anyone ever made it more than four turns without picking the wrong compass direction and getting swallowed by some instantly fatal (and apparently very stealthy) red mist.
The first game I played outside of school (may have been before the BBC Micro experience) was Breakout on my friend's ZX Spectrum.
-Nick