People have been trying to forecast the weather for thousands of years with varying degrees of success. Folksy stories about a green sky meaning a tornado may occasionally "work", but until mankind had a proper understanding of the what the air was doing, and until instruments were invented to measure them, little real progress would take place.
Ancient Greece saw the first attempts to document a manual of weather forecasting. It was they who gave us the word Meteorology. However, they lacked a way to obtain an accurate picture of the weather over a large area, or to accurately measure. The books written were a curious mixture of local truths (when the wind blows from the sea it is cooler) and a lot of mysticism. So they were doomed to disappoint.
It wasn't until weather instruments were developed in the Middle Ages that weather forecasting and observation began to make progress.
The thermometer, as the telescope, is credited to Galileo Galilei. Galileo used alcohol in his thermometer. Mercury was introduced about a hundred years later by Gabriel Fahrenheit in 1709. Since there was no reference standard, early thermometers lacked stable values for anchor points. Various attempts included using the blood temperatures of animals, of melted butter, or that of a hot summer day. The trouble with all of these choices is they were not dependable.
A more precise and unchanging scale came by way of the Swedish astronomer Anders Celsius in 1742, with the invention of the scale that bears his name. This scale (which at one time was called the "centigrade" scale because of 100 points between freezing and boiling), is the accepted worldwide standard for meteorological measurements.
A companion instrument to the thermometer and one even more important for weather forecasting is the barometer. Evangelista Torricelli, a professor of mathematics at the Florentine Academy (Italy), developed the liquid-filled barometer in 1643.
Pressure tendency was found to relate to weather changes: falling pressure is usually accompanied by unsettled or stormy conditions while rising pressure is most often associated with "nice" weather. This makes the barometer crucial is tracking high and low pressure zones.
The anemometer, a wind speed measuring device, is often credited to an English scientist named Robert Hooke. His primitive version used a hanging metal place attached to a dial with evenly spaced markings. The stronger the wind blew, the further the plate swung out and the higher the wind speed reading would be. This type of elementary wind measuring device can still be found for sale with cheap back-yard weather kits. They are not very accurate, but better than nothing.
These three devices, along with companion studies by other atmospheric scientists, established the basic framework for weather observation and reporting still used today.
In the year 1819, Wilhelm Brandes gathered 365 old observations from a European weather pattern of 1783 to plot the first "modern" weather map. Obviously, his map was of no use for that day's weather prediction, but it opened the door to an understanding of pressure systems and their movement. It was an important step forward in the young science of meteorology.
The linking together of weather stations by telegraph in the late 1800s allowed the first crude "modern" surface weather network, and daily map creation. The addition of weather balloons in the 1940s to retrieve upper air information led to the first successful tornado forecast in March of 1948 at Tinker Air Force base (Oklahoma).
Computer models (see here for a detailed description) have also greatly furthered our ability to predict the weather out several days. Complex math needed to make the models work was impossible to "crunch" fast enough by hand. High-speed information processing finally became available starting in the 1960s and the models have leapfrogged in accuracy with each decade since.
More recent advances such as Doppler weather radar make short-range forecasting of severe weather possible in a way not dreamed of when the Greeks penned their thoughts two millenia ago.