Weather and climate are most often predicted using numerical models.
These models start from a snapshot of the current atmospheric conditions in the area of interest, from the surface to the upper atmosphere, at points on a three-dimensional grid. A set of atmospheric variables, such as the wind speed, temperature, pressure and humidity in each grid box are then stored and a set of equations are solved for each grid box to predict the values at that point a short time later. This process is repeated many times; each time the forecast stepping a few minutes further into the future to produce either a weather forecast of the next few days or a climate prediction of the coming 100 years.
The set of equations which are solved fall broadly into 2 categories: the dynamical core solves the equations of motion for a fluid, on a rotating sphere, to calculate the evolution of the atmospheric flow. Alongside the dynamical core, a large number of other Physical Processes operate to warm/cool or moisten/dry the atmosphere, form clouds and precipitation and represent both the weather which we experience, and the effect of that weather on the evolution of the atmospheric flow.