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First, some terminology. These are all basic concepts for computer modeling of the weather, but since you probably don't have much experience yet on computer modeling of the weather, this will all be foreign to you.
Horizontal discretization - The method by which horizontal variations in the atmosphere are treated. The most common method is a regularly-spaced grid, as described in the numerical weather prediction module. The atmospheric conditions at each grid point are kept track of, and are a function of the weather at surrounding grid points. An alternative involves decomposing the horizontal variations into waves of varying amplitude and wavelength. The difference in the two types of horizontal discretization does not matter much to you as a forecaster.
Vertical coordinate - All models use discrete vertical levels, just like horizontal grid points. The obvious choice for a vertical coordinate, height or pressure, is not in common use because it is numerically difficult to cope with a sloping lower boundary. Instead, the models generally use some variation of sigma coordinates (named after the greek letter), with the lowest level following the Earth's surface and higher model levels defined as being some fraction of distance (in pressure) from the ground to the top of the atmosphere.
The vertical coordinate surfaces get flatter and flatter as you get higher and higher into the atmosphere.
Horizontal resolution - The distance between grid points. The finer (smaller) the resolution, the more accurately the model can simulate smaller-scale atmospheric features.
Vertical resolution - Analogous to horizontal resolution. Again, the smaller the resolution, the better the forecast from a given model. All forecast models have finer resolution near the ground, to better simulate the interactions of the atmosphere with the Earth's surface.
Domain - The horizontal extent of the atmosphere simulated by the model. The domain may be global, in which case the whole troposphere and lower stratosphere is simulated, or limited-area, in which case only a portion of the Earth's atmosphere is simulated. Limited-area models use forecasts from a global model to specify the evolution of weather on the edges (boundaries) of the domain.
Update frequency - How often a new set of model forecasts is made.
Analysis cycle - A very important aspect of numerical forecasting is the analysis. If you don't start off with the correct initial conditions, the forecast will start out wrong and only get worse. Most models construct their analysis by performing a series of short forecasts, three to six hours long, and using each forecast as the basis for a new analysis.
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