What is the place called where the next batter up stands?

it depends on if your left handed or right handed and how fast or slow the pitcher is throwing

if you are right handed while facing the pitcher you stand on the left side of the batters box. right side if you are left handed.

if it is a fast pitcher stand toward the back of the box to give yourself more time to read the pitch and swing at the ball. slower pitcher towards the front of the box

if you are bunting stand in the front so if the ball goes straight down it wont hit the plate