To answer the question directly: "style" basically describes what sorts of positions you prefer. If you prefer closed positions with a slow manouevering game you're a "positional" player. If you prefer open positions with lots of tactics all over the place, you're a "tactical" player.
But really, it's not as important as some make it out to be. People say that Tal was a tactical player and Petrosian was a positional player; but Tal understood positional play and Petrosian could execute stunning tactical combinations. Masters are masters because they've mastered all aspects of chess. We are not masters because we haven't mastered the game. So why worry about our style? There's lots of other things we have to learn first.
I am curious as to how one would go about determining their playing style.
I once heard on here that a person does not have a playing style until they reach a certain rating. Is this true?
To be honest, I don't even know what the different playing styles are.