Once we have the current weather data from around the globe, we run the data through some incredibly complex mathematical formulas for how we know the atmosphere works, and then extrapolate that data over the next 1-10 days.
It'll then display the forecasted areas of clouds, temperature, humidity, wind patterns, etc. over whatever area we decide -- models can range from global in scale to as close as just western Washington.
Once you have that data, the trick is then interpreting the data into a generalized forecast. Most models can also take that data, and apply it to what's happened in similar situations in the past, and give out forecasted temperatures, rainfall amounts, humidity, etc.