Weather forecasting is actually a very new science.  Most of what we know now has been learned in the last 100 years. Advances in weather prediction came during the 20th century due to the use of computers. Using computers to predict weather is call "Numerical" weather prediction.

In 1922, Lewis Fry Richardson advanced a new theory that weather prediction could be possible by calculating the results of equations, primarily physics equations that apply to the motion of fluids. This theory became the basis for using computers to predict large scale movements of air masses. Today, there are hundreds of weather models that run on super computers. This output is used to predict motions of weather systems. While there have been many advancements, we can still only accurately predict weather about 5-7 days in advance. However, some scientists believe that as faster computers are developed and more observations are made available as input, that we will inevitably be able to predict weather weeks and months in advance. This is often referred to as the "Computer Problem" or "Data Problem" with forecasting weather.