Snapchat is an app made so you can only see photos for a set period of time (and NOT save them). A screenshot is when you take and save an image of what is on the phone/ipod screen at any given time (by pressing home button and power button simultaneously). It is difficult to do in snapchat, but it is possible. The app will inform you if the person you sent the image to you took a screenshot (and therefore has the image you sent saved on their device). Be careful with that app.
It is when the person u send the snap to takes a screen shot by clicking the top and hime buttons down on an iphone (idk about other phones) this photo will show up in their photo library and can be viewed longer than the allotted 10 seconds