The simplest way is to simply add in a Page Viewer web part, and use this for a the link:
http://www.bing.com/webslices/answer.aspx?q=weather&a=WeatherAnswer&s=Weather&form=IE8WSV
That will display the weather for the current user's general IP address area. You can customize the link to include specific zip codes, if you like, by chaing the "q=weather" to "q=weather XXXXX", where XXXXX is the zip code.
The next simplest is to use a Content Editor Web Part, and choose any of the weather gadgets on Google Gadgets. Once you find one you like in the Google Gadget area, click on "Get the Code", and then copy and pase the code into the "Source Editor..." area of the Content Editor Web Part.
If you would like a finished, free solution, you can use the Bamboo Solutions weather web part that they donated to the Community Kit for SharePoint project.
I'd post a link to that project, but the built-in spam protection here limits me to one link per answer. Do a google search for CKS Weather, and you will get the link from the first hit.