In the Rubin causal model, a potential outcome of a variable $Y$ is simply “the value that $Y$ would have taken for individual $u$, had $X$ been assigned the value $x$.” It’s often convenient to write this quantity more compactly as $Y_X = _x(u)$. Often we abbreviate this further as $Y_x(u)$ if it is apparent from the context what variable is being set to the value x.