What is Manhattan routing?

Manhattan routing is a PCB routing strategy. You use one dedicated layer for horizontal tracks and another layer for vertical tracks. No horizontal tracks are allowed on the vertical layer, and no vertical traces are used on the horizontal layer. This means that most connections will go trough a via, but this strategy can provide surprisingly dense boards with little routing effort.

Look at a map of Manhattan: streets are straight and at right angles. That's what you do on your PCB: horizontal and vertical lines.

There's also the concept of "Manhattan distance". That's the distance measured vertically, then horizontally, instead of a direct line.