The science of weather prediction or forecasting is called meteorology. A person who predicts the weather is a meteorologist.
A meteorologist will collect data from a range of sources to show what the current weather conditions are. They then predict what will happen based on what has happened in the past, when the conditions were the same or similar.
Among the type of data collected is air pressure, temperature, wind speed and direction. This data is collected by weather stations which are dotted around the country.
A video explanation: