NowCoast is a free real-time web mapping portal to information from the National Oceanic and Atmospheric Administration (NOAA). The web mapping application is good and there are plenty of layers to choose from. There are even WMS servers to tap into here: http://nowcoast.noaa.gov/help/mapservices.shtml?name=mapservices. The National Weather Service (NWS), whose parent agency is NOAA, also has a page separated by filetype: shapefiles, KML/KMZs, and web services. A screenshot of nowCOAST appears below.
The Weather Underground's WunderMap, also free, provides a number of features and layers.
Intellicast also has an interactive weather map. Many weather websites also offer low cost subscription plans that upgrade the amount of features, data, and maps you can visualize. Some sites are fee-based, but also provide free trials.
If you are more interested in climate and long-term trends be sure to check out the National Climate Data Centers (NCDC) Interactive Data Online Page. Other resources are your local or state meterologist office, often affiliated with a university.
In addition, a number of very low cost and free mobile weather and radar apps are also available for Android and iOS devices. Lastly, if you are not satisfied with your current weather website, check out this listing of 150 weather sites for North America.
Update #1: Be sure to check out this recent blog post on the NWS Enhanced data viewer: