situation for the umpire to make a ruling on what is  yours: 

Runner on first base, batter with a 2-2 count.  The pitches made in the batter swings at the pitch and tips the ball so that it goes directly into the catchers chest protector and then drops into his glove. The runner  takes off the second and a throw was made in the runner called out.  The coach protest at the runner is not out  sent back to first base.

Now the question is is the right and why, and then what would we do with the batter. =================================================

now hopefully before you read this answer you thought about the question. The answer is it's a foul ball any ball that does not go directly into the catcher's glove is considered a foul ball. And all the actions undertaken when a foul balls made are taken in this case so the runner goes back to first, and foul balls call on the batter so the count stays 2-2.  If it had been a foul -tip  directly into the catcher's glove and in the play proceeded as described the runner would've been out in the batter would have been out on strike three. So it's important to know the difference between a foul ball in a foul tip.