Weather is "the atmospheric conditions at a certain time or over a certain short period in a given area." The challenges in describing the weather on Jupiter are its enormous size and the distance from which we are attempting to analyze those atmospheric conditions.
The driving force in determining the weather on any planet is heat. As the gases in the atmosphere are heated, they move and circulate, expanding toward colder areas. In Jupiter's case, the main source of heat is the core of the planet itself. As the gases in Jupiter's atmosphere are heated, they form huge storm systems that appear to move horizontally around Jupiter due to its rapid rotation. The Great Red Spot is an enormous storm - so large that it absorbs other storms into itself, so large it has existed for hundreds of years.
The temperature on Jupiter depends on location. In the outer reaches of the gaseous atmosphere and on the side that is away from the Sun at a given point in time, the temperature is near absolute zero, the coldest possible temperature. Close to the core of the planet, the temperature may reach 25,000 degrees Kelvin.
The weather conditions on Jupiter vary greatly from place to place, depending upon elevation and location in relation to storm systems, atmospheric pressure conditions, and the Sun.