The atmosphere is a fluid. It obeys the laws of fluid mechanics, thermodynamics, and conservation of mass and water. If we can solve these equations, then we can forecast the weather.
Unfortunately, no one has yet found an analytical solution to these equations. Instead, we can approximate these continuous differential equations with finite-difference equations (i.e., algebra) that we can solve on a computer. One approach is to divide the spatial domain into finite-sized grid cells, and to forecast the average conditions at a grid point within each cell.
If we know (or can approximate) the initial condition of each grid point by assimilating new observations into past forecasts, then we can make an iterative forecast by taking finite-size time steps into the future. To do this, all grid points must be marched together one step at a time.
Some aspects of atmospheric physics cause resolvable changes to the forecast, even though the grid resolution is not fine enough to resolve all the physical details. Hence, physical processes such as turbulence, radiation, clouds, and precipitation must be parameterized as a simplified function of variables that can be resolved in the model (winds, temperature, etc.).
The finite-difference equations suffer from truncation, round-off, numerical instability, and dynamic instability errors. Round-off errors are smaller when more bits are used to represent numbers in a computer. Truncation errors are smaller when more of the higher-order terms are retained in Taylor-series approximations of derivatives. Numerical instability is reduced when the time step is sufficiently small relative to the grid size. Dynamic instability refers to the sensitive dependence of the forecast on initial conditions and model parameters. Dynamic instability can be reduced with better weather analyses, but it cannot be eliminated.
No numerical forecast is perfect. For any specific location, forecasts might have a consistent bias or systematic error. Most of these biases can be removed by using statistics such as model output statistics (MOS) to post-process the NWP output. Random errors associated with the chaotic nonlinearly dynamic nature of the atmosphere can be estimated and/or reduced by making multiple forecasts (called ensemble forecasts) with different initial conditions or parameterizations. The ensemble forecasts can be averaged to give a deterministic forecast, and they can be used to make probabilistic forecasts.
Unfortunately, we have not discovered a way to reduce all errors. Thus, forecast error usually increases with increasing forecast time (i.e., how far into the future you forecast). Forecast skill is often defined relative to some baseline or reference, such as climatology. Short-range (out to 3 days) forecasts show significant skill, while medium-range forecasts show modest skill out to about 10 days. In addition to deterministic forecasts, NWP is increasingly being used to make probabilistic forecasts. Methods exist to verify deterministic, binary, and probabilistic forecasts.
The resulting forecasts can be analyzed and graphed to reveal cyclones, fronts, airmasses, and other weather systems. Of all the tools (satellites, radar, weather balloons, etc.) that meteorologists use to forecast the weather, only NWP gives the future weather with a skill that is better than persistence.