On a staggered grid the scalar variables (pressure, density, total enthalpy etc.) are stored in the cell centers of the control volumes, whereas the velocity or momentum variables are located at the cell faces. This is different from a collocated grid arrangement, where all variables are stored in the same positions. A staggered storage is mainly used on structured grids for compressible or incompressible flow simulations.
Using a staggered grid is a simple way to avoid odd-even decoupling between the pressure and velocity. Odd-even decoupling is a discretization error that can occur on collocated grids and which leads to checkerboard patterns in the solutions.
The disadvantage of using staggered grids is that different variable are stored at different places and this makes it more difficult to handle different control volumes for different varibales and to keep track of the metrics. Most modern codes instead use a collocated storage.
Harlow, F. H. and Welch, J. E. (1965), "Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface", Phys. Fluids. 8, 2182.