As it happens, NASA has an 8 part article about this very subject.

Basically weather forecasting was made possible on a local level with the advent of the thermometer, the barometer and the hygrometer; all of which had been invented by about the mid 17th century. With these tools local observers could study weather in their area. However it took the advent of the telegraph to allow communication between meteorologists over a great enough distance to make large scale weather maps. These early attempts were rudimentary and often subject to technical failure.

In the 1920's a device known as the radiosonde, launched via weather balloons, transmitted weather data to stations which could be used to predict weather patterns over a large area. The problem is that this data took vast amounts of time and people to calculate a forecast. Lewis Fry Richardson was a pioneer of the mathematical weather forecast, but it would take computer technology to calculate the data fast enough to make a useful forecast. This was accomplished by the mid 1950's.

On April 1st, 1960 the satellite TIROS 1 gave us the first pictures of weather systems from orbit, and with that data accurate weather observation and prediction was finally possible.