I think you're confusing "run" and "home run". In high-level terms:
- A run is scored when a player starts off as a batter then makes it all the way back round to home plate without being put out, no matter if he stops at any bases along the way.
- A home run happens when a batter hits the ball over the fences (or equivalent). When this happens, any players currently on first, second or third bases also get to finish their circle of the bases and score a run (but not a home run).
As such in your situation, two runs are scored: one for the player who was already on first, and one for the batter who hit the home run. However, only the batter is credited with a home run. It's possibly also worth noting here that for the purposes of winning the game, runs are the only thing that matters. It doesn't matter if they were home runs or not - they all count as runs; it's perfectly possible to win a game without scoring a single home run, but you obviously have to score at least one run.