Manhattan distance, aka city block distance, is the number of units one must traverse to get from point X to point Y along a grid-like path. In other words, each step involves a change in value across exactly one dimension. In machine learning, Manhattan distance is preferred (over higher values of p) for high-dimensionality data because of simpler computation and because of something called the "curse of high dimensionality."

More formally, let $x_i$ be the value of point X across the $i$th dimension, and let $y_i$ be the value of point Y across the $i$th dimension (NOT to be confused with the conventional (x,y) notation for a point in 2D). Then, in $n$ dimensions, the Manhattan distance between points X and Y is $d(X,Y) = \sum_{i=1}^n|x_i - y_i|$.

(Note that this is an instantiation of the Minkowski distance formula with p=1.)