What is the study of the weather called?

Meteorology, and the name has nothing to do with meteors. It comes from Aristotle's Meteorologica, in which meteoron meant anything high in the air - clouds, rain, rainbows and, incidentally, shooting stars, which were assumed to be an atmospheric phenomenon. When the rocks turned out to come from space they kept the name and the weather kept it too.

Forecasting became possible with the telegraph, because a forecast requires knowing what is happening upwind faster than the weather can travel. Robert FitzRoy - who had captained the Beagle with Darwin aboard - set up the first storm-warning service in Britain in the 1860s and coined the word forecast. He was mocked for it, and the criticism contributed to his suicide.

The reason forecasts fail is one of the founding results of chaos theory. Edward Lorenz, running a weather simulation in 1961, restarted it from a printout rounded to three decimal places instead of six and got a completely different result. Tiny differences in the starting state grow exponentially, which puts a hard ceiling of about two weeks on any weather prediction, however good the model. The talk he gave about it is where the butterfly effect comes from.