Left-Handed Hitters, Right-Handed Hitters, and Switch Hitters
A player's dominant hand determines how he grips the bat and on which side of home plate he stands. Players grip the bat with their dominant hand on top, and stand with their non-dominant shoulder closer to the pitcher. Batters tend to hit better off pitchers who have the opposite dominant hand as them, because the pitch enters their vision sooner.
Switch hitters are players who can bat on either side of the plate. They usually are not ambidextrous themselves, they have just been trained to hit both left-handed and right-handed. Being a switch hitter is useful because they can hit against both left-handed and right-handed pitchers; there is no disadvantage to hitting against a pitcher with the same dominant hand because they can simply bat from the other side of the plate.
Square Stance, Open Stance, and Closed Stance
The square batting stance is the most common stance. The batter stands with both his feet an equal distance from the plate, and parallel to the side of the batter's box. It allows batters to see the pitcher with both eyes, and to be in a good, balanced position to hit any pitch.
The open batting stance is when the batter stands with his front foot farther from home plate than his back foot. This causes his chest to be facing the pitcher slightly, making him "open" to the pitcher. It allows batters to track the pitch better, and to pull the ball.
The closed batting stance is when the batter stands with his back foot farther home plate than his front foot. Since this causes his back to be facing the pitcher slightly, he is "closed" to the pitcher. Although this stance makes it harder to see the ball, some batters claim it helps them hit the ball more powerfully.