Probably the most common thing that comes to mind when discussion keeping score is hits and errors. Rule 10.12a1 explains that an error is a “fumble, muff, or wild throw” that prolongs the time of a batter at the plate, the time a runner is on base, or allows a runner to advance a base.
It reminds us that a slow handling of a ball is not grounds for an error—that is to say a ground ball to the shortstop that is fielded cleanly and accurately thrown to first base is not an error if the runner beats it out.
The rule goes on to explain that an error is only to be charged if the official scorer deems that the play should’ve been made by a player using “ordinary effort.” No third baseman is expected to charge in, field a bunt with his bare hand, and make an off balance throw to get a base runner out a first by a step.
You should also factor in the level of play. No little leaguer is expected to be Ichiro.
The only way to get a good feel for hits and errors and what is and is not “ordinary effort” is to watch as much ball as possible. A helpful resource are the videos on MLB.com; obviously all of the plays are on a Major League level, but the scorekeepers are the best in the world at what they do.