The National Weather Service has drastically improved its app looking web site for mobile devices. To access it, go to mobile.weather.gov on any smart phone browser. Once you are there, open up the browser's menu and select "Add to homescreen" or something similar. The web site functions like an app, yet there is no need to download or install anything.
Here are some screen shots showing how this site looks, feels, and works. First, the site comes up showing the current conditions and a simple forecast for a default location. You can change the default location and add other locations by zip codes, city name, or by touching a point on a map: