A wall kick happens when a player rotates a piece when no space exists in the squares where that tetromino would normally occupy after the rotation. To compensate, the game sets a certain number of alternative spaces for the tetromino to look. Wall kicks started appearing in games because after a piece would be backed up against the wall, it would awkwardly be unable to rotate until first moving back.
Here T can rotate
Here T cannot rotate without a wall kick
The simplest wall kick algorithm, used (with variations) by TGM rotation and several fan games in the same tradition, is to try moving the tetromino one space to the right, and then one space to the left, and fail if neither can be done. Wall kicks increase the number of possible twists.