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Weather and climate models do not represent the atmosphere in a continuous manner, but either at discrete points or averages around those points. This is the case in the horizontal (east-west and north-south), vertical and in time.
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.
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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.
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