A tatter is a raggedy end or scrap of something, particularly paper or fabric. Your crazy cat might climb the curtains every night, until there's nothing left hanging from the rod but tatters.
It's most common to find this word in its plural form, tatters. A homeless person might dress in tatters, and a serious hiker might wear a pair of socks until they're just tatters, then throw them away and put on a new pair. Tatter comes from an earlier word, tatrys, "slashed garments," which has a Scandinavian root and is related to words like the Old Norse töturr, "rags."