Ubuntu ships with a weather indicator application. But, there has been at least one computer I've used on which it would just not accept any city as input to show its weather. Also, you may be interested in seeing more weather statistics than the simple ones shown by the default weather application.
My Weather Indicator is a good alternate weather application and indicator for Ubuntu. It shows far more statistics, for example dawn and dusk times and many more like that. It is also far more customizable with many settings.
Tried with: My Weather Indicator 0.6.3 and Ubuntu 12.04 LTS