You can save any picture or video you take in Snapchat to your iPhone, iPad Android, Nexus, Windows phone, or any other compatible device. Follow these steps.
1. Open the Snapchat app.
2. Take a video or picture. At the bottom of the same screen, you will see an icon to save your media to your phone camera photo/video archive.
3. Once saved, Snapchat will let you know.
iPhone: Photos App -> Snapchat
Android / Nexus: Galley App