Alpine Museum, Zermatt
This is an ideal visit if the weather is bad or if you are taking a break from a nice day of hiking. The museum focuses on several aspects of life in the mountains. It takes into account the geological composition, plant life (especially the wildflowers), some of the native animals, and the stories of some of the famous mountain climbers as they went after the big peaks. Among the exhibits is the ice axe of the first person to climb the Matterhorn. There is an admission charge, although it is free to those with a valid Swiss Pass.
Only open for a few hours a day, so make sure you plan ahead. I think it was open between 4pm and 6pm when we were there. They are also moving to a new building before 2006.