Weather and climate models do not represent the atmosphere in a continuous manner, but either at discrete points or averages around those points.
In general, models cannot represent spatial variability on scales smaller than those defined by the spacing between the discrete points, the grid scale. Model values described as representing values at the discrete points, actually represent some sort of average value over the grid scale. However, models don't represent the grid scale very well either, with the effective resolution of models being somewhat larger than the grid scale.
Horizontal
In the horizontal the discrete points are arranged in a two dimensional grid and hence are referred to as grid points. The grid can be regular or irregular. An example of a regular latitude/longitude grid would be where the grid points are located every 1 degree of longitude in the east-west direction and every 1 degree of latitude in the north-south direction. Each grid point is associated with an area that either surrounds the grid point or lies between the grid points. This area is referred to as the "grid box".
Grid point values cannot properly represent variability on spatial scales smaller than the grid box.
(In addition to using grid points, the ECMWF Integrated Forecasting System (IFS) also uses an additional mathematical concept, spectral space, to represent horizontal space. This concept uses a set of wavy basis functions, spherical harmonics, to describe variations in the horizontal. The IFS switches between spectral space and grid point space, in order to perform specific computations.)
A description of the grid used for ERA5 data is given at:
ERA5: What is the spatial reference
A description of the grid used for ERA-Interim data is given at:
ERA-Interim: What is the spatial reference
Vertical
In the vertical, models can use levels, located at discrete points, and/or averages over layers.
Time
Weather and climate models usually represent time at discrete points, with a spacing that is called the time step. The time step typically varies between a few minutes and half an hour, depending on the model and its configuration.
Model parameters cannot represent variability on short time scales, but tend to evolve smoothly, for more information see the article "Parameters valid at the specified time".
