We have forecast data from several regional models that cover United States. These models have a higher resolution than global models that forecast for the whole world, but the increased computational demands of a high resolution model mean that these forecasts only go a couple days out in time, and are limited to a small area.
We have forecast data from a variety of global models that produce forecasts for the whole world, up to two weeks out in time. These models are all generally fairly accurate in predicting large scale patterns/features, but all will become less accurate through time. The ECMWF is generally considered to be the most accurate global model, with the US's GFS slightly behind.
Weather models, known formally as "Numerical Weather Prediction" are at the core of modern weather forecasts. All the forecast information you see at weather.us is powered by weather models, do what are they and how do they work?
Weather models are simulations of the future state of the atmosphere out through time. Millions of observations are used as initial conditions in trillions of calculations, producing a three dimensional picture of what the atmosphere might look like at some time in the future. Massive computers are used to do these calculations at incredibly fast speeds to enable simulations to cover the entire globe, and extend up to two weeks into the future.
There are two general types of weather models, global models and regional models. Global models produce forecast output for the whole globe, generally extending a week or two into the future. Because these models cover a wider area, and a longer timespan, they’re generally run at a lower resolution, both spatially (fewer forecast points per given area) and temporally (fewer time points get a forecast).
Regional models on the other hand have much higher resolutions, but only cover some part (region) of the globe, and only provide forecasts a couple days out in time. The advantage with these models is that their higher resolution lets them "see" features that the global models miss, most notably including thunderstorms.
Many different national weather centers have supercomputers that run weather models. Each of these is slightly different, using different equations to solve for various physical processes that shape our weather patterns. Many of them also have slightly different resolutions, and use slightly different combinations of initial data sources.
These slight differences multiply out through time because the atmosphere is a chaotic system. This also means any errors that the models make in the near term become exponentially larger with time. This is why the forecast for a week from now is far less accurate than the forecast for tomorrow.
Weather modelling centers attempt to control for the influence of chaos by running ensemble systems that each use slightly different initial conditions. Each ensemble "member" then produces a forecast as if its set of initial conditions were correct. This provides some way of quantifying how likely a given forecast outcome is, helping to show forecast uncertainty.