Here are some examples of horizontal grids used in ESMs from around the web. A simple grid is the longitude-latitude (or lon-lat) grid, such as the one shown here:

Geo-referencing (i.e mapping to actual points on the planetary surface) is usually done with respect to geographic latitude and longitude, and so the lon-lat grid can be thought of as an underlying canonical grid for Earth system studies. Many multi-model studies (e. IPCC AR4) have required data to be placed on this grid before being accepted to a public archive.

But models might use more specialized grids from various algorithmic considerations. Examples include the tripolar and cubed-sphere grids, more extensively discussed in the draft Gridspec:

as well as others, even more exotic:

The vertical is generally separated out because the dynamical scales involved differ by many orders of magnitude from the horizontal, and the physics is quite different. And it's further complicated by the fact that the lower boundary (the planetary surface) is a complex topographic map. Additionally, can be good reasons for choosing a vertical coordinate that follows material surfaces (e.g isolines of pressure or density). Here's an illustration showing the difference between a pressure coordinate and a terrain-following σ-coordinate:

In the case of ocean models, an additional consideration is the undulating upper free surface. Vertical coordinates have been developed in this case that not only respect the bottom terrain, but the free surface as well: