We don't think much about weather on other planets, but any world with an atmosphere has weather. Earth's weather is unique because the chemistry and physics of our atmosphere allows water to exist as a gas, liquid, and solid all at once.
The Martian atmosphere has the right temperatures and pressures, but only enough water for very thin clouds. Mars does have windstorms and even dust devils, some of which can reach heights of 10 miles.
Venus has thick clouds made of carbon dioxide, but virtually no water. A typical day on Venus would be cloudy with a temperature near 800 degrees.
Jupiter, Saturn, Uranus and Neptune all have methane-based atmospheres and tremendous air pressures which produce bizarre carbon-based precipitation, essentially liquid and solid diamonds falling through the air. Jupiter's Great Red Spot is a giant, circulating windstorm in the Jovian upper atmosphere that has been slowly shrinking for decades.
Mercury has no atmosphere and, therefore, no weather. Any weather that might exist on Pluto remains mostly a mystery.
